Bernoulli number - Wikipedia

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href="#Integral_Expression"> <div class="vector-toc-text"> 3.4 Integral Expression </div> </a> <ul id="toc-Integral_Expression-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bernoulli_numbers_and_the_Riemann_zeta_function" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bernoulli_numbers_and_the_Riemann_zeta_function"> <div class="vector-toc-text"> 4 Bernoulli numbers and the Riemann zeta function </div> </a> <ul id="toc-Bernoulli_numbers_and_the_Riemann_zeta_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Efficient_computation_of_Bernoulli_numbers" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Efficient_computation_of_Bernoulli_numbers"> <div class="vector-toc-text"> 5 Efficient computation of Bernoulli numbers </div> </a> <ul id="toc-Efficient_computation_of_Bernoulli_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications_of_the_Bernoulli_numbers" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications_of_the_Bernoulli_numbers"> <div class="vector-toc-text"> 6 Applications of the Bernoulli numbers </div> </a> <button aria-controls="toc-Applications_of_the_Bernoulli_numbers-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications of the Bernoulli numbers subsection </button> <ul id="toc-Applications_of_the_Bernoulli_numbers-sublist" class="vector-toc-list"> <li id="toc-Asymptotic_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Asymptotic_analysis"> <div class="vector-toc-text"> 6.1 Asymptotic analysis </div> </a> <ul id="toc-Asymptotic_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sum_of_powers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sum_of_powers"> <div class="vector-toc-text"> 6.2 Sum of powers </div> </a> <ul id="toc-Sum_of_powers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Taylor_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Taylor_series"> <div class="vector-toc-text"> 6.3 Taylor series </div> </a> <ul id="toc-Taylor_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Laurent_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Laurent_series"> <div class="vector-toc-text"> 6.4 Laurent series </div> </a> <ul id="toc-Laurent_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Use_in_topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Use_in_topology"> <div class="vector-toc-text"> 6.5 Use in topology </div> </a> <ul id="toc-Use_in_topology-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Connections_with_combinatorial_numbers" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Connections_with_combinatorial_numbers"> <div class="vector-toc-text"> 7 Connections with combinatorial numbers </div> </a> <button aria-controls="toc-Connections_with_combinatorial_numbers-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Connections with combinatorial numbers subsection </button> <ul id="toc-Connections_with_combinatorial_numbers-sublist" class="vector-toc-list"> <li id="toc-Connection_with_Worpitzky_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connection_with_Worpitzky_numbers"> <div class="vector-toc-text"> 7.1 Connection with Worpitzky numbers </div> </a> <ul id="toc-Connection_with_Worpitzky_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_with_Stirling_numbers_of_the_second_kind" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connection_with_Stirling_numbers_of_the_second_kind"> <div class="vector-toc-text"> 7.2 Connection with Stirling numbers of the second kind </div> </a> <ul id="toc-Connection_with_Stirling_numbers_of_the_second_kind-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_with_Stirling_numbers_of_the_first_kind" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connection_with_Stirling_numbers_of_the_first_kind"> <div class="vector-toc-text"> 7.3 Connection with Stirling numbers of the first kind </div> </a> <ul id="toc-Connection_with_Stirling_numbers_of_the_first_kind-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_with_Pascal's_triangle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connection_with_Pascal's_triangle"> <div class="vector-toc-text"> 7.4 Connection with Pascal's triangle </div> </a> <ul id="toc-Connection_with_Pascal's_triangle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_with_Eulerian_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connection_with_Eulerian_numbers"> <div class="vector-toc-text"> 7.5 Connection with Eulerian numbers </div> </a> <ul id="toc-Connection_with_Eulerian_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-A_binary_tree_representation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#A_binary_tree_representation"> <div class="vector-toc-text"> 8 A binary tree representation </div> </a> <ul id="toc-A_binary_tree_representation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integral_representation_and_continuation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Integral_representation_and_continuation"> <div class="vector-toc-text"> 9 Integral representation and continuation </div> </a> <ul id="toc-Integral_representation_and_continuation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_relation_to_the_Euler_numbers_and_π" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#The_relation_to_the_Euler_numbers_and_π"> <div class="vector-toc-text"> 10 The relation to the Euler numbers and π </div> </a> <ul id="toc-The_relation_to_the_Euler_numbers_and_π-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-An_algorithmic_view:_the_Seidel_triangle" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#An_algorithmic_view:_the_Seidel_triangle"> <div class="vector-toc-text"> 11 An algorithmic view: the Seidel triangle </div> </a> <ul id="toc-An_algorithmic_view:_the_Seidel_triangle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_combinatorial_view:_alternating_permutations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#A_combinatorial_view:_alternating_permutations"> <div class="vector-toc-text"> 12 A combinatorial view: alternating permutations </div> </a> <ul id="toc-A_combinatorial_view:_alternating_permutations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_sequences" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Related_sequences"> <div class="vector-toc-text"> 13 Related sequences </div> </a> <ul id="toc-Related_sequences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arithmetical_properties_of_the_Bernoulli_numbers" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Arithmetical_properties_of_the_Bernoulli_numbers"> <div class="vector-toc-text"> 14 Arithmetical properties of the Bernoulli numbers </div> </a> <button aria-controls="toc-Arithmetical_properties_of_the_Bernoulli_numbers-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Arithmetical properties of the Bernoulli numbers subsection </button> <ul id="toc-Arithmetical_properties_of_the_Bernoulli_numbers-sublist" class="vector-toc-list"> <li id="toc-The_Kummer_theorems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Kummer_theorems"> <div class="vector-toc-text"> 14.1 The Kummer theorems </div> </a> <ul id="toc-The_Kummer_theorems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-p-adic_continuity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#p-adic_continuity"> <div class="vector-toc-text"> 14.2 p-adic continuity </div> </a> <ul id="toc-p-adic_continuity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ramanujan's_congruences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ramanujan's_congruences"> <div class="vector-toc-text"> 14.3 Ramanujan's congruences </div> </a> <ul id="toc-Ramanujan's_congruences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Von_Staudt–Clausen_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Von_Staudt–Clausen_theorem"> <div class="vector-toc-text"> 14.4 Von Staudt–Clausen theorem </div> </a> <ul id="toc-Von_Staudt–Clausen_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Why_do_the_odd_Bernoulli_numbers_vanish?" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Why_do_the_odd_Bernoulli_numbers_vanish?"> <div class="vector-toc-text"> 14.5 Why do the odd Bernoulli numbers vanish? </div> </a> <ul id="toc-Why_do_the_odd_Bernoulli_numbers_vanish?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_restatement_of_the_Riemann_hypothesis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_restatement_of_the_Riemann_hypothesis"> <div class="vector-toc-text"> 14.6 A restatement of the Riemann hypothesis </div> </a> <ul id="toc-A_restatement_of_the_Riemann_hypothesis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalized_Bernoulli_numbers" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalized_Bernoulli_numbers"> <div class="vector-toc-text"> 15 Generalized Bernoulli numbers </div> </a> <button aria-controls="toc-Generalized_Bernoulli_numbers-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Generalized Bernoulli numbers subsection </button> <ul id="toc-Generalized_Bernoulli_numbers-sublist" class="vector-toc-list"> <li id="toc-Eisenstein–Kronecker_number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eisenstein–Kronecker_number"> <div class="vector-toc-text"> 15.1 Eisenstein–Kronecker number </div> </a> <ul id="toc-Eisenstein–Kronecker_number-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Appendix" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Appendix"> <div class="vector-toc-text"> 16 Appendix </div> </a> <button aria-controls="toc-Appendix-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Appendix subsection </button> <ul id="toc-Appendix-sublist" class="vector-toc-list"> <li id="toc-Assorted_identities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Assorted_identities"> <div class="vector-toc-text"> 16.1 Assorted identities </div> </a> <ul id="toc-Assorted_identities-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 17 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 18 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 19 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> 20 Bibliography </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 21 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Bernoulli number</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" 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Available in 36 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-36" aria-hidden="true" > 36 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%A8%D8%B1%D9%86%D9%88%D9%84%D9%8A" title="عدد برنولي – Arabic" lang="ar" hreflang="ar" data-title="عدد برنولي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Bernulli_%C9%99d%C9%99dl%C9%99ri" title="Bernulli ədədləri – Azerbaijani" lang="az" hreflang="az" data-title="Bernulli ədədləri" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%BD%D0%B0_%D0%91%D0%B5%D1%80%D0%BD%D1%83%D0%BB%D0%B8" title="Числа на Бернули – Bulgarian" lang="bg" hreflang="bg" data-title="Числа на Бернули" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombres_de_Bernoulli" title="Nombres de Bernoulli – Catalan" lang="ca" hreflang="ca" data-title="Nombres de Bernoulli" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bernoulliho_%C4%8D%C3%ADslo" title="Bernoulliho číslo – Czech" lang="cs" hreflang="cs" data-title="Bernoulliho číslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Bernoulli-Zahl" title="Bernoulli-Zahl – German" lang="de" hreflang="de" data-title="Bernoulli-Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82_%CE%9C%CF%80%CE%B5%CF%81%CE%BD%CE%BF%CF%8D%CE%BB%CE%B9" title="Αριθμός Μπερνούλι – Greek" lang="el" hreflang="el" data-title="Αριθμός Μπερνούλι" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_de_Bernoulli" title="Número de Bernoulli – Spanish" lang="es" hreflang="es" data-title="Número de Bernoulli" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bernoulliren_zenbaki" title="Bernoulliren zenbaki – Basque" lang="eu" hreflang="eu" data-title="Bernoulliren zenbaki" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%A8%D8%B1%D9%86%D9%88%D9%84%DB%8C" title="عدد برنولی – Persian" lang="fa" hreflang="fa" data-title="عدد برنولی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_de_Bernoulli" title="Nombre de Bernoulli – French" lang="fr" hreflang="fr" data-title="Nombre de Bernoulli" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B2%A0%EB%A5%B4%EB%88%84%EC%9D%B4_%EC%88%98" title="베르누이 수 – Korean" lang="ko" hreflang="ko" data-title="베르누이 수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A4%B0%E0%A5%8D%E0%A4%A8%E0%A5%82%E0%A4%B2%E0%A5%80_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="बर्नूली संख्या – Hindi" lang="hi" hreflang="hi" data-title="बर्नूली संख्या" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numeri_di_Bernoulli" title="Numeri di Bernoulli – Italian" lang="it" hreflang="it" data-title="Numeri di Bernoulli" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8%D7%99_%D7%91%D7%A8%D7%A0%D7%95%D7%9C%D7%99" title="מספרי ברנולי – Hebrew" lang="he" hreflang="he" data-title="מספרי ברנולי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D0%B5%D1%80%D0%BD%D1%83%D0%BB%D0%BB%D0%B8_%D1%81%D0%B0%D0%BD%D0%B4%D0%B0%D1%80%D1%8B" title="Бернулли сандары – Kazakh" lang="kk" hreflang="kk" data-title="Бернулли сандары" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Bernulli_skaitlis" title="Bernulli skaitlis – Latvian" lang="lv" hreflang="lv" data-title="Bernulli skaitlis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Bernoulli-sz%C3%A1mok" title="Bernoulli-számok – Hungarian" lang="hu" hreflang="hu" data-title="Bernoulli-számok" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Bernoulligetal" title="Bernoulligetal – Dutch" lang="nl" hreflang="nl" data-title="Bernoulligetal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%99%E3%83%AB%E3%83%8C%E3%83%BC%E3%82%A4%E6%95%B0" title="ベルヌーイ数 – Japanese" lang="ja" hreflang="ja" data-title="ベルヌーイ数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bernoulli-tall" title="Bernoulli-tall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Bernoulli-tall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Bernoulli_sonlari" title="Bernoulli sonlari – Uzbek" lang="uz" hreflang="uz" data-title="Bernoulli sonlari" 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clear:right; margin-left:1em;"> <caption>Bernoulli numbers B± n </caption> <tbody><tr> <th>n</th> <th>fraction</th> <th>decimal </th></tr> <tr> <td>0</td> <td>1</td> <td>+1.000000000 </td></tr> <tr> <td>1</td> <td>±<style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠1/2⁠</td> <td>±0.500000000 </td></tr> <tr> <td>2</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠</td> <td>+0.166666666 </td></tr> <tr style="background:#ABE"> <td>3</td> <td>0</td> <td>+0.000000000 </td></tr> <tr> <td>4</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/30⁠</td> <td>−0.033333333 </td></tr> <tr style="background:#ABE"> <td>5</td> <td>0</td> <td>+0.000000000 </td></tr> <tr> <td>6</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/42⁠</td> <td>+0.023809523 </td></tr> <tr style="background:#ABE"> <td>7</td> <td>0</td> <td>+0.000000000 </td></tr> <tr> <td>8</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/30⁠</td> <td>−0.033333333 </td></tr> <tr style="background:#ABE"> <td>9</td> <td>0</td> <td>+0.000000000 </td></tr> <tr> <td>10</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/66⁠</td> <td>+0.075757575 </td></tr> <tr style="background:#ABE"> <td>11</td> <td>0</td> <td>+0.000000000 </td></tr> <tr> <td>12</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠691/2730⁠</td> <td>−0.253113553 </td></tr> <tr style="background:#ABE"> <td>13</td> <td>0</td> <td>+0.000000000 </td></tr> <tr> <td>14</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠7/6⁠</td> <td>+1.166666666 </td></tr> <tr style="background:#ABE"> <td>15</td> <td>0</td> <td>+0.000000000 </td></tr> <tr> <td>16</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3617/510⁠</td> <td>−7.092156862 </td></tr> <tr style="background:#ABE"> <td>17</td> <td>0</td> <td>+0.000000000 </td></tr> <tr> <td>18</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠43867/798⁠</td> <td>+54.97117794 </td></tr> <tr style="background:#ABE"> <td>19</td> <td>0</td> <td>+0.000000000 </td></tr> <tr> <td>20</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠174611/330⁠</td> <td>−529.1242424 </td></tr></tbody></table> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the Bernoulli numbers Bn are a <a href="/wiki/Sequence" title="Sequence">sequence</a> of <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> which occur frequently in <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a>. The Bernoulli numbers appear in (and can be defined by) the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> expansions of the <a href="/wiki/Tangent_function" class="mw-redirect" title="Tangent function">tangent</a> and <a href="/wiki/Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic tangent</a> functions, in <a href="/wiki/Faulhaber%27s_formula" title="Faulhaber's formula">Faulhaber's formula</a> for the sum of m-th powers of the first n positive integers, in the <a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a>, and in expressions for certain values of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}^{-{}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}^{-{}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/329eac7d7c4716537965a031529735c7708382d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.275ex; height:2.843ex;" alt="{\displaystyle B_{n}^{-{}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}^{+{}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}^{+{}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55a5c0445044c7b4b59ae151e6e7a8c0a0886558" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.275ex; height:2.843ex;" alt="{\displaystyle B_{n}^{+{}}}">; they differ only for n = 1, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}^{-{}}=-1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}^{-{}}=-1/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c266ffff482a015334b3e66fc8a746317ec270f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.669ex; height:3.176ex;" alt="{\displaystyle B_{1}^{-{}}=-1/2}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}^{+{}}=+1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> </msubsup> <mo>=</mo> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}^{+{}}=+1/2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcf6d1e4bcd3ba88ff33c5ceb1e3756fc963b506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.669ex; height:3.176ex;" alt="{\displaystyle B_{1}^{+{}}=+1/2}">. For every odd n > 1, Bn = 0. For every even n > 0, Bn is negative if n is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the <a href="/wiki/Bernoulli_polynomials" title="Bernoulli polynomials">Bernoulli polynomials</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9192cc7ff70d2e7aff04305da16f00c76a42e1bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.121ex; height:2.843ex;" alt="{\displaystyle B_{n}(x)}">, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}^{-{}}=B_{n}(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> </msubsup> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}^{-{}}=B_{n}(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f0b194fc4859d8462731b5811c1d742efb37c14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.328ex; height:3.009ex;" alt="{\displaystyle B_{n}^{-{}}=B_{n}(0)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}^{+}=B_{n}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}^{+}=B_{n}(1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a929d3e5b9c7b9056b395d49e9f78e5331a1e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.328ex; height:3.009ex;" alt="{\displaystyle B_{n}^{+}=B_{n}(1)}">.<a href="#cite_note-Weisstein2016-1">[1]</a> The Bernoulli numbers were discovered around the same time by the Swiss mathematician <a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a>, after whom they are named, and independently by Japanese mathematician <a href="/wiki/Seki_Takakazu" title="Seki Takakazu">Seki Takakazu</a>. Seki's discovery was posthumously published in 1712<a href="#cite_note-Selin1997_891-2">[2]</a><a href="#cite_note-SmithMikami1914_108-3">[3]</a><a href="#cite_note-Kitagawa-4">[4]</a> in his work Katsuyō Sanpō; Bernoulli's, also posthumously, in his <a href="/wiki/Ars_Conjectandi" title="Ars Conjectandi">Ars Conjectandi</a> of 1713. <a href="/wiki/Ada_Lovelace" title="Ada Lovelace">Ada Lovelace</a>'s <a href="/wiki/Note_G" title="Note G">note G</a> on the <a href="/wiki/Analytical_Engine" class="mw-redirect" title="Analytical Engine">Analytical Engine</a> from 1842 describes an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> for generating Bernoulli numbers with <a href="/wiki/Charles_Babbage" title="Charles Babbage">Babbage</a>'s machine;<a href="#cite_note-Menabrea1842_noteG-5">[5]</a> it is disputed <a href="/wiki/Ada_Lovelace#Controversy_over_contribution" title="Ada Lovelace">whether Lovelace or Babbage developed the algorithm</a>. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex <a href="/wiki/Computer_program" title="Computer program">computer program</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=1" title="Edit section: Notation">edit</a>]</div> The superscript ± used in this article distinguishes the two sign conventions for Bernoulli numbers. Only the n = 1 term is affected: <ul><li>B− n with B− 1 = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ (<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A027641" class="extiw" title="oeis:A027641">A027641</a> / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A027642" class="extiw" title="oeis:A027642">A027642</a>) is the sign convention prescribed by <a href="/wiki/NIST" class="mw-redirect" title="NIST">NIST</a> and most modern textbooks.<a href="#cite_note-FOOTNOTEArfken1970278-6">[6]</a></li> <li>B+ n with B+ 1 = +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ (<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A164555" class="extiw" title="oeis:A164555">A164555</a> / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A027642" class="extiw" title="oeis:A027642">A027642</a>) was used in the older literature,<a href="#cite_note-Weisstein2016-1">[1]</a> and (since 2022) by <a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a><a href="#cite_note-7">[7]</a> following Peter Luschny's "Bernoulli Manifesto".<a href="#cite_note-8">[8]</a></li></ul> In the formulas below, one can switch from one sign convention to the other with the relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}^{+}=(-1)^{n}B_{n}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}^{+}=(-1)^{n}B_{n}^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc22fd5f2d42d2020de196bae82f11b77c07ed1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.646ex; height:3.009ex;" alt="{\displaystyle B_{n}^{+}=(-1)^{n}B_{n}^{-}}">, or for integer n = 2 or greater, simply ignore it. Since Bn = 0 for all odd n > 1, and many formulas only involve even-index Bernoulli numbers, a few authors write "Bn" instead of B2n . This article does not follow that notation. <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=2" title="Edit section: History">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Early_history">Early history</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=3" title="Edit section: Early history">edit</a>]</div> The Bernoulli numbers are rooted in the early history of the computation of sums of integer powers, which have been of interest to mathematicians since antiquity. <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png/220px-Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png" decoding="async" width="220" height="290" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png/330px-Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png/440px-Seki_Kowa_Katsuyo_Sampo_Bernoulli_numbers.png 2x" data-file-width="1614" data-file-height="2124" /></a><figcaption>A page from Seki Takakazu's Katsuyō Sanpō (1712), tabulating binomial coefficients and Bernoulli numbers</figcaption></figure> Methods to calculate the sum of the first n positive integers, the sum of the squares and of the cubes of the first n positive integers were known, but there were no real 'formulas', only descriptions given entirely in words. Among the great mathematicians of antiquity to consider this problem were <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a> (c. 572–497 BCE, Greece), <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> (287–212 BCE, Italy), <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a> (b. 476, India), <a href="/wiki/Abu_Bakr_al-Karaji" class="mw-redirect" title="Abu Bakr al-Karaji">Abu Bakr al-Karaji</a> (d. 1019, Persia) and Abu Ali al-Hasan ibn al-Hasan ibn <a href="/wiki/Al-Haytham" class="mw-redirect" title="Al-Haytham">al-Haytham</a> (965–1039, Iraq). During the late sixteenth and early seventeenth centuries mathematicians made significant progress. In the West <a href="/wiki/Thomas_Harriot" title="Thomas Harriot">Thomas Harriot</a> (1560–1621) of England, <a href="/wiki/Johann_Faulhaber" title="Johann Faulhaber">Johann Faulhaber</a> (1580–1635) of Germany, <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> (1601–1665) and fellow French mathematician <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a> (1623–1662) all played important roles. Thomas Harriot seems to have been the first to derive and write formulas for sums of powers using symbolic notation, but even he calculated only up to the sum of the fourth powers. Johann Faulhaber gave formulas for sums of powers up to the 17th power in his 1631 Academia Algebrae, far higher than anyone before him, but he did not give a general formula. Blaise Pascal in 1654 proved <a href="/wiki/Faulhaber%27s_formula" title="Faulhaber's formula">Pascal's identity</a> relating the sums of the pth powers of the first n positive integers for p = 0, 1, 2, ..., k. The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence of constants B0, B1, B2,... which provide a uniform formula for all sums of powers.<a href="#cite_note-FOOTNOTEKnuth1993-9">[9]</a> The joy Bernoulli experienced when he hit upon the pattern needed to compute quickly and easily the coefficients of his formula for the sum of the cth powers for any positive integer c can be seen from his comment. He wrote: <dl><dd>"With the help of this table, it took me less than half of a quarter of an hour to find that the tenth powers of the first 1000 numbers being added together will yield the sum 91,409,924,241,424,243,424,241,924,242,500."</dd></dl> Bernoulli's result was published posthumously in <a href="/wiki/Ars_Conjectandi" title="Ars Conjectandi">Ars Conjectandi</a> in 1713. <a href="/wiki/Seki_Takakazu" title="Seki Takakazu">Seki Takakazu</a> independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712.<a href="#cite_note-Selin1997_891-2">[2]</a> However, Seki did not present his method as a formula based on a sequence of constants. Bernoulli's formula for sums of powers is the most useful and generalizable formulation to date. The coefficients in Bernoulli's formula are now called Bernoulli numbers, following a suggestion of <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a>. Bernoulli's formula is sometimes called <a href="/wiki/Faulhaber%27s_formula" title="Faulhaber's formula">Faulhaber's formula</a> after Johann Faulhaber who found remarkable ways to calculate sum of powers but never stated Bernoulli's formula. According to Knuth<a href="#cite_note-FOOTNOTEKnuth1993-9">[9]</a> a rigorous proof of Faulhaber's formula was first published by <a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Carl Jacobi</a> in 1834.<a href="#cite_note-Jacobi1834-10">[10]</a> Knuth's in-depth study of Faulhaber's formula concludes (the nonstandard notation on the LHS is explained further on): <dl><dd>"Faulhaber never discovered the Bernoulli numbers; i.e., he never realized that a single sequence of constants B0, B1, B2, ... would provide a uniform <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum n^{m}={\frac {1}{m+1}}\left(B_{0}n^{m+1}-{\binom {m+1}{1}}B_{1}n^{m}+{\binom {m+1}{2}}B_{2}n^{m-1}-\cdots +(-1)^{m}{\binom {m+1}{m}}B_{m}n\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mo>⋯</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>n</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum n^{m}={\frac {1}{m+1}}\left(B_{0}n^{m+1}-{\binom {m+1}{1}}B_{1}n^{m}+{\binom {m+1}{2}}B_{2}n^{m-1}-\cdots +(-1)^{m}{\binom {m+1}{m}}B_{m}n\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d4984d9288fee26abd5b1056cc7fffc014b2c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:80.163ex; height:3.676ex;" alt="{\textstyle \sum n^{m}={\frac {1}{m+1}}\left(B_{0}n^{m+1}-{\binom {m+1}{1}}B_{1}n^{m}+{\binom {m+1}{2}}B_{2}n^{m-1}-\cdots +(-1)^{m}{\binom {m+1}{m}}B_{m}n\right)}"></dd></dl></dd> <dd>for all sums of powers. He never mentioned, for example, the fact that almost half of the coefficients turned out to be zero after he had converted his formulas for Σ nm from polynomials in N to polynomials in n."<a href="#cite_note-FOOTNOTEKnuth199314-11">[11]</a></dd></dl> In the above Knuth meant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b92c3380e64a08b439589e4c3407b09ba625676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.275ex; height:3.176ex;" alt="{\displaystyle B_{1}^{-}}">; instead using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caa61e42fdddd902ee7eddb6f1a211dbf95864b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.275ex; height:3.176ex;" alt="{\displaystyle B_{1}^{+}}"> the formula avoids subtraction: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum n^{m}={\frac {1}{m+1}}\left(B_{0}n^{m+1}+{\binom {m+1}{1}}B_{1}^{+}n^{m}+{\binom {m+1}{2}}B_{2}n^{m-1}+\cdots +{\binom {m+1}{m}}B_{m}n\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum n^{m}={\frac {1}{m+1}}\left(B_{0}n^{m+1}+{\binom {m+1}{1}}B_{1}^{+}n^{m}+{\binom {m+1}{2}}B_{2}n^{m-1}+\cdots +{\binom {m+1}{m}}B_{m}n\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6373388302cd9ee514fb2474fb3a03ad80fee23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:75.199ex; height:3.676ex;" alt="{\textstyle \sum n^{m}={\frac {1}{m+1}}\left(B_{0}n^{m+1}+{\binom {m+1}{1}}B_{1}^{+}n^{m}+{\binom {m+1}{2}}B_{2}n^{m-1}+\cdots +{\binom {m+1}{m}}B_{m}n\right).}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Reconstruction_of_"Summae_Potestatum"">Reconstruction of "Summae Potestatum"</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=4" title="Edit section: Reconstruction of "Summae Potestatum"">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:JakobBernoulliSummaePotestatum.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/JakobBernoulliSummaePotestatum.png/330px-JakobBernoulliSummaePotestatum.png" decoding="async" width="330" height="419" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/JakobBernoulliSummaePotestatum.png/495px-JakobBernoulliSummaePotestatum.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/74/JakobBernoulliSummaePotestatum.png 2x" data-file-width="576" data-file-height="732" /></a><figcaption>Jakob Bernoulli's "Summae Potestatum", 1713<a href="#cite_note-12">[a]</a></figcaption></figure> The Bernoulli numbers <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A164555" class="extiw" title="oeis:A164555">A164555</a>(n)/<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A027642" class="extiw" title="oeis:A027642">A027642</a>(n) were introduced by Jakob Bernoulli in the book <a href="/wiki/Ars_Conjectandi" title="Ars Conjectandi">Ars Conjectandi</a> published posthumously in 1713 page 97. The main formula can be seen in the second half of the corresponding facsimile. The constant coefficients denoted A, B, C and D by Bernoulli are mapped to the notation which is now prevalent as A = B2, B = B4, C = B6, D = B8. The expression c·c−1·c−2·c−3 means c·(c−1)·(c−2)·(c−3) – the small dots are used as grouping symbols. Using today's terminology these expressions are <a href="/wiki/Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">falling factorial powers</a> ck. The factorial notation k! as a shortcut for 1 × 2 × ... × k was not introduced until 100 years later. The integral symbol on the left hand side goes back to <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> in 1675 who used it as a long letter S for "summa" (sum).<a href="#cite_note-13">[b]</a> The letter n on the left hand side is not an index of <a href="/wiki/Summation" title="Summation">summation</a> but gives the upper limit of the range of summation which is to be understood as 1, 2, ..., n. Putting things together, for positive c, today a mathematician is likely to write Bernoulli's formula as: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}k^{c}={\frac {n^{c+1}}{c+1}}+{\frac {1}{2}}n^{c}+\sum _{k=2}^{c}{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mi>c</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>_</mo> </munder> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}k^{c}={\frac {n^{c+1}}{c+1}}+{\frac {1}{2}}n^{c}+\sum _{k=2}^{c}{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ad3bc96c357dfa3baafe1119960dba553a7693e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.71ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}k^{c}={\frac {n^{c+1}}{c+1}}+{\frac {1}{2}}n^{c}+\sum _{k=2}^{c}{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}.}"></dd></dl> This formula suggests setting B1 = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ when switching from the so-called 'archaic' enumeration which uses only the even indices 2, 4, 6... to the modern form (more on different conventions in the next paragraph). Most striking in this context is the fact that the <a href="/wiki/Falling_factorial#Real_numbers_and_negative_n" class="mw-redirect" title="Falling factorial">falling factorial</a> ck−1 has for k = 0 the value <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/c + 1⁠.<a href="#cite_note-FOOTNOTEGrahamKnuthPatashnik1989Section_2.51-14">[12]</a> Thus Bernoulli's formula can be written <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}k^{c}=\sum _{k=0}^{c}{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>_</mo> </munder> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}k^{c}=\sum _{k=0}^{c}{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94800378ba11b922cadd6795c732684350c35bc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.199ex; height:7.009ex;" alt="{\displaystyle \sum _{k=1}^{n}k^{c}=\sum _{k=0}^{c}{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}}"></dd></dl> if B1 = 1/2, recapturing the value Bernoulli gave to the coefficient at that position. The formula for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{k=1}^{n}k^{9}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{k=1}^{n}k^{9}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6083bca54a40c95a8df6812c6901238f95ead91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.296ex; height:3.176ex;" alt="{\displaystyle \textstyle \sum _{k=1}^{n}k^{9}}"> in the first half of the quotation by Bernoulli above contains an error at the last term; it should be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\tfrac {3}{20}}n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {3}{20}}n^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e38b33d36fcb0e951ebf4d0c922e28d21dc8ae2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.737ex; height:3.676ex;" alt="{\displaystyle -{\tfrac {3}{20}}n^{2}}"> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\tfrac {1}{12}}n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mstyle> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{12}}n^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca96e1525b32a6766b1977134543c0f397db6871" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.737ex; height:3.509ex;" alt="{\displaystyle -{\tfrac {1}{12}}n^{2}}">. <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=5" title="Edit section: Definitions">edit</a>]</div> Many characterizations of the Bernoulli numbers have been found in the last 300 years, and each could be used to introduce these numbers. Here only four of the most useful ones are mentioned: <ul><li>a recursive equation,</li> <li>an explicit formula,</li> <li>a generating function,</li> <li>an integral expression.</li></ul> For the proof of the <a href="/wiki/Logical_equivalence" title="Logical equivalence">equivalence</a> of the four approaches.<a href="#cite_note-15">[13]</a> <div class="mw-heading mw-heading3"><h3 id="Recursive_definition">Recursive definition</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=6" title="Edit section: Recursive definition">edit</a>]</div> The Bernoulli numbers obey the sum formulas<a href="#cite_note-Weisstein2016-1">[1]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sum _{k=0}^{m}{\binom {m+1}{k}}B_{k}^{-{}}&=\delta _{m,0}\\\sum _{k=0}^{m}{\binom {m+1}{k}}B_{k}^{+{}}&=m+1\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{k=0}^{m}{\binom {m+1}{k}}B_{k}^{-{}}&=\delta _{m,0}\\\sum _{k=0}^{m}{\binom {m+1}{k}}B_{k}^{+{}}&=m+1\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05f4218423d2c0287c23de640d572ae231647835" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:26.374ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\sum _{k=0}^{m}{\binom {m+1}{k}}B_{k}^{-{}}&=\delta _{m,0}\\\sum _{k=0}^{m}{\binom {m+1}{k}}B_{k}^{+{}}&=m+1\end{aligned}}}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=0,1,2...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=0,1,2...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16e0a6d44602dd78f055b7849a7325e5479e3313" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.635ex; height:2.509ex;" alt="{\displaystyle m=0,1,2...}"> and δ denotes the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>. The first of these is sometimes written<a href="#cite_note-16">[14]</a> as the formula (for m > 1) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (B+1)^{m}-B_{m}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B+1)^{m}-B_{m}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ff540d6c6a910f3310a97bc5f1a8cf50ea9c981" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.438ex; height:2.843ex;" alt="{\displaystyle (B+1)^{m}-B_{m}=0,}"> where the power is expanded formally using the binomial theorem and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f1ebbc79c2f3fe878d51107b5497de8f3c964c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.853ex; height:2.676ex;" alt="{\displaystyle B^{k}}"> is replaced by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6457760e36cf45e1471e33bcc1536cb4802fb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.853ex; height:2.509ex;" alt="{\displaystyle B_{k}}">. Solving for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{m}^{\mp {}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∓</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{m}^{\mp {}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4ae8520e36a9804c722ae23587fda18d499168" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.439ex; height:2.843ex;" alt="{\displaystyle B_{m}^{\mp {}}}"> gives the recursive formulas<a href="#cite_note-17">[15]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}B_{m}^{-{}}&=\delta _{m,0}-\sum _{k=0}^{m-1}{\binom {m}{k}}{\frac {B_{k}^{-{}}}{m-k+1}}\\B_{m}^{+}&=1-\sum _{k=0}^{m-1}{\binom {m}{k}}{\frac {B_{k}^{+}}{m-k+1}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>m</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> </msubsup> <mrow> <mi>m</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>m</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mrow> <mi>m</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}B_{m}^{-{}}&=\delta _{m,0}-\sum _{k=0}^{m-1}{\binom {m}{k}}{\frac {B_{k}^{-{}}}{m-k+1}}\\B_{m}^{+}&=1-\sum _{k=0}^{m-1}{\binom {m}{k}}{\frac {B_{k}^{+}}{m-k+1}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/393561628c724ec2226e4934ef4ba6f67388d3a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:34.439ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}B_{m}^{-{}}&=\delta _{m,0}-\sum _{k=0}^{m-1}{\binom {m}{k}}{\frac {B_{k}^{-{}}}{m-k+1}}\\B_{m}^{+}&=1-\sum _{k=0}^{m-1}{\binom {m}{k}}{\frac {B_{k}^{+}}{m-k+1}}.\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Explicit_definition">Explicit definition</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=7" title="Edit section: Explicit definition">edit</a>]</div> In 1893 <a href="/wiki/Louis_Saalsch%C3%BCtz" title="Louis Saalschütz">Louis Saalschütz</a> listed a total of 38 explicit formulas for the Bernoulli numbers,<a href="#cite_note-Saalschütz1893-18">[16]</a> usually giving some reference in the older literature. One of them is (for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e0f3243e9d7f06bee548558bf20aaa9b5263d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.301ex; height:2.343ex;" alt="{\displaystyle m\geq 1}">): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}B_{m}^{-}&=\sum _{k=0}^{m}{\frac {1}{k+1}}\sum _{j=0}^{k}{\binom {k}{j}}(-1)^{j}j^{m}\\B_{m}^{+}&=\sum _{k=0}^{m}{\frac {1}{k+1}}\sum _{j=0}^{k}{\binom {k}{j}}(-1)^{j}(j+1)^{m}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}B_{m}^{-}&=\sum _{k=0}^{m}{\frac {1}{k+1}}\sum _{j=0}^{k}{\binom {k}{j}}(-1)^{j}j^{m}\\B_{m}^{+}&=\sum _{k=0}^{m}{\frac {1}{k+1}}\sum _{j=0}^{k}{\binom {k}{j}}(-1)^{j}(j+1)^{m}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/429c0d02ac41dab739a43044de9edef1478ee7db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:40.625ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}B_{m}^{-}&=\sum _{k=0}^{m}{\frac {1}{k+1}}\sum _{j=0}^{k}{\binom {k}{j}}(-1)^{j}j^{m}\\B_{m}^{+}&=\sum _{k=0}^{m}{\frac {1}{k+1}}\sum _{j=0}^{k}{\binom {k}{j}}(-1)^{j}(j+1)^{m}.\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Generating_function">Generating function</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=8" title="Edit section: Generating function">edit</a>]</div> The exponential <a href="/wiki/Generating_function" title="Generating function">generating functions</a> are <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{3}{\frac {t}{e^{t}-1}}&={\frac {t}{2}}\left(\operatorname {coth} {\frac {t}{2}}-1\right)&&=\sum _{m=0}^{\infty }{\frac {B_{m}^{-{}}t^{m}}{m!}}\\{\frac {te^{t}}{e^{t}-1}}={\frac {t}{1-e^{-t}}}&={\frac {t}{2}}\left(\operatorname {coth} {\frac {t}{2}}+1\right)&&=\sum _{m=0}^{\infty }{\frac {B_{m}^{+}t^{m}}{m!}}.\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>coth</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> </msubsup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>t</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>coth</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{3}{\frac {t}{e^{t}-1}}&={\frac {t}{2}}\left(\operatorname {coth} {\frac {t}{2}}-1\right)&&=\sum _{m=0}^{\infty }{\frac {B_{m}^{-{}}t^{m}}{m!}}\\{\frac {te^{t}}{e^{t}-1}}={\frac {t}{1-e^{-t}}}&={\frac {t}{2}}\left(\operatorname {coth} {\frac {t}{2}}+1\right)&&=\sum _{m=0}^{\infty }{\frac {B_{m}^{+}t^{m}}{m!}}.\end{alignedat}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2446067113eeb75dd9ba80c3c36e130d7a7464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:52.777ex; height:14.176ex;" alt="{\displaystyle {\begin{alignedat}{3}{\frac {t}{e^{t}-1}}&={\frac {t}{2}}\left(\operatorname {coth} {\frac {t}{2}}-1\right)&&=\sum _{m=0}^{\infty }{\frac {B_{m}^{-{}}t^{m}}{m!}}\\{\frac {te^{t}}{e^{t}-1}}={\frac {t}{1-e^{-t}}}&={\frac {t}{2}}\left(\operatorname {coth} {\frac {t}{2}}+1\right)&&=\sum _{m=0}^{\infty }{\frac {B_{m}^{+}t^{m}}{m!}}.\end{alignedat}}}"></dd></dl> where the substitution is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\to -t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">→</mo> <mo>−</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\to -t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a6a9c7f4ef3bc3301e225656972f6ab91bb157" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.101ex; height:2.176ex;" alt="{\displaystyle t\to -t}">. The two generating functions only differ by t. <style data-mw-deduplicate="TemplateStyles:r1256386598">.mw-parser-output .cot-header-mainspace{background:#F0F2F5;color:inherit}.mw-parser-output .cot-header-other{background:#CCFFCC;color:inherit}@media screen{html.skin-theme-clientpref-night .mw-parser-output .cot-header-mainspace{background:#14181F;color:inherit}html.skin-theme-clientpref-night .mw-parser-output .cot-header-other{background:#003500;color:inherit}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cot-header-mainspace{background:#14181F;color:inherit}html.skin-theme-clientpref-os .mw-parser-output .cot-header-other{background:#003500;color:inherit}}</style> <div style="margin-left:0"> <table class="mw-collapsible mw-archivedtalk mw-collapsed" style="color:inherit; background: transparent; text-align: left; border: 1px solid Silver; margin: 0.2em auto auto; width:100%; clear: both; padding: 1px;"> <tbody><tr> <th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Proof</div> </th></tr> <tr> <td style="color:inherit; border: solid 1px Silver; padding: 0.6em; background: var(--background-color-base, #fff);"> If we let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(t)=\sum _{i=1}^{\infty }f_{i}t^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(t)=\sum _{i=1}^{\infty }f_{i}t^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c3622f9ce456eee2504e1cd761797cd93e2c16f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.808ex; height:6.843ex;" alt="{\displaystyle F(t)=\sum _{i=1}^{\infty }f_{i}t^{i}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(t)=1/(1+F(t))=\sum _{i=0}^{\infty }g_{i}t^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(t)=1/(1+F(t))=\sum _{i=0}^{\infty }g_{i}t^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dfa7ee2012543540c48a0270b3f65ef771e7f98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.489ex; height:6.843ex;" alt="{\displaystyle G(t)=1/(1+F(t))=\sum _{i=0}^{\infty }g_{i}t^{i}}"> then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(t)=1-F(t)G(t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(t)=1-F(t)G(t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35821d44261847d4354cad46ddd5fb4cf843537e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.089ex; height:2.843ex;" alt="{\displaystyle G(t)=1-F(t)G(t).}"></dd></dl> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{0}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8adcdbc326a0270e3ceb5e0bce18652111dff107" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.424ex; height:2.509ex;" alt="{\displaystyle g_{0}=1}"> and for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/501173910e6da8425b4e9d44a4e8643620bc2464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m>0}"> the mth term in the series for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3d6c09ba5569413364689bf4837c7b71ef0892f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.476ex; height:2.843ex;" alt="{\displaystyle G(t)}"> is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{m}t^{i}=-\sum _{j=0}^{m-1}f_{m-j}g_{j}t^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mi>j</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{m}t^{i}=-\sum _{j=0}^{m-1}f_{m-j}g_{j}t^{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/467b7b9c5813faa714aefd6deb425cb0d2217740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.951ex; height:7.676ex;" alt="{\displaystyle g_{m}t^{i}=-\sum _{j=0}^{m-1}f_{m-j}g_{j}t^{m}}"></dd></dl> If <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(t)={\frac {e^{t}-1}{t}}-1=\sum _{i=1}^{\infty }{\frac {t^{i}}{(i+1)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mi>t</mi> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(t)={\frac {e^{t}-1}{t}}-1=\sum _{i=1}^{\infty }{\frac {t^{i}}{(i+1)!}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fdfd689b32216d6074a318434dc23729a26193a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.178ex; height:6.843ex;" alt="{\displaystyle F(t)={\frac {e^{t}-1}{t}}-1=\sum _{i=1}^{\infty }{\frac {t^{i}}{(i+1)!}}}"></dd></dl> then we find that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(t)=t/(e^{t}-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(t)=t/(e^{t}-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8adbe534b37be2983c5df051d9aef8a70c52b811" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.298ex; height:3.009ex;" alt="{\displaystyle G(t)=t/(e^{t}-1)}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}m!g_{m}&=-\sum _{j=0}^{m-1}{\frac {m!}{j!}}{\frac {j!g_{j}}{(m-j+1)!}}\\&=-{\frac {1}{m+1}}\sum _{j=0}^{m-1}{\binom {m+1}{j}}j!g_{j}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>m</mi> <mo>!</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo>!</mo> </mrow> <mrow> <mi>j</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>j</mi> <mo>!</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mi>j</mi> <mo>!</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}m!g_{m}&=-\sum _{j=0}^{m-1}{\frac {m!}{j!}}{\frac {j!g_{j}}{(m-j+1)!}}\\&=-{\frac {1}{m+1}}\sum _{j=0}^{m-1}{\binom {m+1}{j}}j!g_{j}\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/324a38bee9dfc06c1350bdc036135c0402fb82c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:35.414ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}m!g_{m}&=-\sum _{j=0}^{m-1}{\frac {m!}{j!}}{\frac {j!g_{j}}{(m-j+1)!}}\\&=-{\frac {1}{m+1}}\sum _{j=0}^{m-1}{\binom {m+1}{j}}j!g_{j}\\\end{aligned}}}"></dd></dl> showing that the values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i!g_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>!</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i!g_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/567c8a05c48b04883b7505adad680e155b2bcbc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.358ex; height:2.509ex;" alt="{\displaystyle i!g_{i}}"> obey the recursive formula for the Bernoulli numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{i}^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{i}^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abef886d008d31a3d123f76ae7a24fd98d3c42e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.275ex; height:3.176ex;" alt="{\displaystyle B_{i}^{-}}">. </td></tr></tbody></table></div> The (ordinary) generating function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{-1}\psi _{1}(z^{-1})=\sum _{m=0}^{\infty }B_{m}^{+}z^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{-1}\psi _{1}(z^{-1})=\sum _{m=0}^{\infty }B_{m}^{+}z^{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55fe3eb95729ad5f8ebd45204faf7b255c2ac944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.456ex; height:6.843ex;" alt="{\displaystyle z^{-1}\psi _{1}(z^{-1})=\sum _{m=0}^{\infty }B_{m}^{+}z^{m}}"></dd></dl> is an <a href="/wiki/Asymptotic_series" class="mw-redirect" title="Asymptotic series">asymptotic series</a>. It contains the <a href="/wiki/Trigamma_function" title="Trigamma function">trigamma function</a> ψ1. <div class="mw-heading mw-heading3"><h3 id="Integral_Expression">Integral Expression</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=9" title="Edit section: Integral Expression">edit</a>]</div> From the generating functions above, one can obtain the following integral formula for the even Bernoulli numbers: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{2n}=4n(-1)^{n+1}\int _{0}^{\infty }{\frac {t^{2n-1}}{e^{2\pi t}-1}}\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mi>n</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2n}=4n(-1)^{n+1}\int _{0}^{\infty }{\frac {t^{2n-1}}{e^{2\pi t}-1}}\mathrm {d} t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48e1e7590847b696c5115ea67be0e73c9ec7e8f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.317ex; height:6.176ex;" alt="{\displaystyle B_{2n}=4n(-1)^{n+1}\int _{0}^{\infty }{\frac {t^{2n-1}}{e^{2\pi t}-1}}\mathrm {d} t}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Bernoulli_numbers_and_the_Riemann_zeta_function">Bernoulli numbers and the Riemann zeta function</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=10" title="Edit section: Bernoulli numbers and the Riemann zeta function">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:BernoulliNumbersByZeta.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/BernoulliNumbersByZeta.svg/220px-BernoulliNumbersByZeta.svg.png" decoding="async" width="220" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/BernoulliNumbersByZeta.svg/330px-BernoulliNumbersByZeta.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/BernoulliNumbersByZeta.svg/440px-BernoulliNumbersByZeta.svg.png 2x" data-file-width="480" data-file-height="288" /></a><figcaption>The Bernoulli numbers as given by the Riemann zeta function.</figcaption></figure> The Bernoulli numbers can be expressed in terms of the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>: <dl><dd>B+ n = −n ζ(1 − n)           for n ≥ 1 .</dd></dl> Here the argument of the zeta function is 0 or negative. As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (k)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ec6e8a5b5544a95f7e2c04134743a6ed0b12772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.116ex; height:2.843ex;" alt="{\displaystyle \zeta (k)}"> is zero for negative even integers (the <a href="/wiki/Riemann_zeta_function#Zeros,_the_critical_line,_and_the_Riemann_hypothesis" title="Riemann zeta function">trivial zeroes</a>), if n>1 is odd, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (1-n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (1-n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ef1996a10814372ae7cdebdc289b4ca1113fe76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.302ex; height:2.843ex;" alt="{\displaystyle \zeta (1-n)}"> is zero. By means of the zeta <a href="/wiki/Riemann_zeta_function#Riemann's_functional_equation" title="Riemann zeta function">functional equation</a> and the gamma <a href="/wiki/Gamma_function#General" title="Gamma function">reflection formula</a> the following relation can be obtained:<a href="#cite_note-FOOTNOTEArfken1970279-19">[17]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{2n}={\frac {(-1)^{n+1}2(2n)!}{(2\pi )^{2n}}}\zeta (2n)\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2n}={\frac {(-1)^{n+1}2(2n)!}{(2\pi )^{2n}}}\zeta (2n)\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5fed8a1d28f2196fc17d7631657be9d35cedb85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.798ex; height:6.676ex;" alt="{\displaystyle B_{2n}={\frac {(-1)^{n+1}2(2n)!}{(2\pi )^{2n}}}\zeta (2n)\quad }"> for n ≥ 1 .</dd></dl> Now the argument of the zeta function is positive. It then follows from ζ → 1 (n → ∞) and <a href="/wiki/Stirling_formula" class="mw-redirect" title="Stirling formula">Stirling's formula</a> that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |B_{2n}|\sim 4{\sqrt {\pi n}}\left({\frac {n}{\pi e}}\right)^{2n}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>∼</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> <mi>n</mi> </msqrt> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |B_{2n}|\sim 4{\sqrt {\pi n}}\left({\frac {n}{\pi e}}\right)^{2n}\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/400c1f40f71fda49f24ede3919b901eac2cab493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.412ex; height:5.176ex;" alt="{\displaystyle |B_{2n}|\sim 4{\sqrt {\pi n}}\left({\frac {n}{\pi e}}\right)^{2n}\quad }"> for n → ∞ .</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Efficient_computation_of_Bernoulli_numbers">Efficient computation of Bernoulli numbers</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=11" title="Edit section: Efficient computation of Bernoulli numbers">edit</a>]</div> In some applications it is useful to be able to compute the Bernoulli numbers B0 through Bp − 3 modulo p, where p is a prime; for example to test whether <a href="/wiki/Vandiver%27s_conjecture" class="mw-redirect" title="Vandiver's conjecture">Vandiver's conjecture</a> holds for p, or even just to determine whether p is an <a href="/wiki/Irregular_prime" class="mw-redirect" title="Irregular prime">irregular prime</a>. It is not feasible to carry out such a computation using the above recursive formulae, since at least (a constant multiple of) p2 arithmetic operations would be required. Fortunately, faster methods have been developed<a href="#cite_note-BuhlerCraErnMetShokrollahi2001-20">[18]</a> which require only O(p (log p)2) operations (see <a href="/wiki/Big-O_notation" class="mw-redirect" title="Big-O notation">big O notation</a>). David Harvey<a href="#cite_note-Harvey2010-21">[19]</a> describes an algorithm for computing Bernoulli numbers by computing Bn modulo p for many small primes p, and then reconstructing Bn via the <a href="/wiki/Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a>. Harvey writes that the <a href="/wiki/Asymptotic_analysis" title="Asymptotic analysis">asymptotic</a> <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">time complexity</a> of this algorithm is O(n2 log(n)2 + ε) and claims that this <a href="/wiki/Implementation" title="Implementation">implementation</a> is significantly faster than implementations based on other methods. Using this implementation Harvey computed Bn for n = 108. Harvey's implementation has been included in <a href="/wiki/SageMath" title="SageMath">SageMath</a> since version 3.1. Prior to that, Bernd Kellner<a href="#cite_note-Kellner2002-22">[20]</a> computed Bn to full precision for n = 106 in December 2002 and Oleksandr Pavlyk<a href="#cite_note-Pavlyk29Apr2008-23">[21]</a> for n = 107 with <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> in April 2008. <style data-mw-deduplicate="TemplateStyles:r1226773818">.mw-parser-output .defaultleft{text-align:left}.mw-parser-output .defaultcenter{text-align:center}.mw-parser-output .defaultright{text-align:right}.mw-parser-output .col1left td:nth-child(1),.mw-parser-output .col2left td:nth-child(2),.mw-parser-output .col3left td:nth-child(3),.mw-parser-output .col4left td:nth-child(4),.mw-parser-output .col5left td:nth-child(5),.mw-parser-output .col6left td:nth-child(6),.mw-parser-output .col7left td:nth-child(7),.mw-parser-output .col8left td:nth-child(8),.mw-parser-output .col9left td:nth-child(9),.mw-parser-output .col10left td:nth-child(10),.mw-parser-output .col11left td:nth-child(11),.mw-parser-output .col12left td:nth-child(12),.mw-parser-output .col13left td:nth-child(13),.mw-parser-output .col14left td:nth-child(14),.mw-parser-output .col15left td:nth-child(15),.mw-parser-output .col16left td:nth-child(16),.mw-parser-output .col17left td:nth-child(17),.mw-parser-output .col18left td:nth-child(18),.mw-parser-output .col19left td:nth-child(19),.mw-parser-output .col20left td:nth-child(20),.mw-parser-output .col21left td:nth-child(21),.mw-parser-output .col22left td:nth-child(22),.mw-parser-output .col23left td:nth-child(23),.mw-parser-output .col24left td:nth-child(24),.mw-parser-output .col25left td:nth-child(25),.mw-parser-output .col26left td:nth-child(26),.mw-parser-output .col27left td:nth-child(27),.mw-parser-output .col28left td:nth-child(28),.mw-parser-output .col29left td:nth-child(29){text-align:left}.mw-parser-output .col1center td:nth-child(1),.mw-parser-output .col2center td:nth-child(2),.mw-parser-output .col3center td:nth-child(3),.mw-parser-output .col4center td:nth-child(4),.mw-parser-output .col5center td:nth-child(5),.mw-parser-output .col6center td:nth-child(6),.mw-parser-output .col7center td:nth-child(7),.mw-parser-output .col8center td:nth-child(8),.mw-parser-output .col9center td:nth-child(9),.mw-parser-output .col10center td:nth-child(10),.mw-parser-output .col11center td:nth-child(11),.mw-parser-output .col12center td:nth-child(12),.mw-parser-output .col13center td:nth-child(13),.mw-parser-output .col14center td:nth-child(14),.mw-parser-output .col15center td:nth-child(15),.mw-parser-output .col16center td:nth-child(16),.mw-parser-output .col17center td:nth-child(17),.mw-parser-output .col18center td:nth-child(18),.mw-parser-output .col19center td:nth-child(19),.mw-parser-output .col20center td:nth-child(20),.mw-parser-output .col21center td:nth-child(21),.mw-parser-output .col22center td:nth-child(22),.mw-parser-output .col23center td:nth-child(23),.mw-parser-output .col24center td:nth-child(24),.mw-parser-output .col25center td:nth-child(25),.mw-parser-output .col26center td:nth-child(26),.mw-parser-output .col27center td:nth-child(27),.mw-parser-output .col28center td:nth-child(28),.mw-parser-output .col29center td:nth-child(29){text-align:center}.mw-parser-output .col1right td:nth-child(1),.mw-parser-output .col2right td:nth-child(2),.mw-parser-output .col3right td:nth-child(3),.mw-parser-output .col4right td:nth-child(4),.mw-parser-output .col5right td:nth-child(5),.mw-parser-output .col6right td:nth-child(6),.mw-parser-output .col7right td:nth-child(7),.mw-parser-output .col8right td:nth-child(8),.mw-parser-output .col9right td:nth-child(9),.mw-parser-output .col10right td:nth-child(10),.mw-parser-output .col11right td:nth-child(11),.mw-parser-output .col12right td:nth-child(12),.mw-parser-output .col13right td:nth-child(13),.mw-parser-output .col14right td:nth-child(14),.mw-parser-output .col15right td:nth-child(15),.mw-parser-output .col16right td:nth-child(16),.mw-parser-output .col17right td:nth-child(17),.mw-parser-output .col18right td:nth-child(18),.mw-parser-output .col19right td:nth-child(19),.mw-parser-output .col20right td:nth-child(20),.mw-parser-output .col21right td:nth-child(21),.mw-parser-output .col22right td:nth-child(22),.mw-parser-output .col23right td:nth-child(23),.mw-parser-output .col24right td:nth-child(24),.mw-parser-output .col25right td:nth-child(25),.mw-parser-output .col26right td:nth-child(26),.mw-parser-output .col27right td:nth-child(27),.mw-parser-output .col28right td:nth-child(28),.mw-parser-output .col29right td:nth-child(29){text-align:right}</style> <dl><dd><table class="wikitable defaultright col1left"> <tbody><tr> <th>Computer</th> <th>Year</th> <th>n</th> <th>Digits* </th></tr> <tr> <td>J. Bernoulli</td> <td>~1689</td> <td>10</td> <td>1 </td></tr> <tr> <td>L. Euler</td> <td>1748</td> <td>30</td> <td>8 </td></tr> <tr> <td>J. C. Adams</td> <td>1878</td> <td>62</td> <td>36 </td></tr> <tr> <td>D. E. Knuth, T. J. Buckholtz</td> <td>1967</td> <td>1672</td> <td>3330 </td></tr> <tr> <td>G. Fee, <a href="/wiki/S._Plouffe" class="mw-redirect" title="S. Plouffe">S. Plouffe</a></td> <td>1996</td> <td>10000</td> <td>27677 </td></tr> <tr> <td>G. Fee, S. Plouffe</td> <td>1996</td> <td>100000</td> <td>376755 </td></tr> <tr> <td>B. C. Kellner</td> <td>2002</td> <td>1000000</td> <td>4767529 </td></tr> <tr> <td>O. Pavlyk</td> <td>2008</td> <td>10000000</td> <td>57675260 </td></tr> <tr> <td>D. Harvey</td> <td>2008</td> <td>100000000</td> <td>676752569 </td></tr></tbody></table></dd></dl> <dl><dd><dl><dd>* Digits is to be understood as the exponent of 10 when Bn is written as a real number in normalized <a href="/wiki/Scientific_notation" title="Scientific notation">scientific notation</a>.</dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Applications_of_the_Bernoulli_numbers">Applications of the Bernoulli numbers</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=12" title="Edit section: Applications of the Bernoulli numbers">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Asymptotic_analysis">Asymptotic analysis</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=13" title="Edit section: Asymptotic analysis">edit</a>]</div> Arguably the most important application of the Bernoulli numbers in mathematics is their use in the <a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a>. Assuming that f is a sufficiently often differentiable function the Euler–Maclaurin formula can be written as<a href="#cite_note-FOOTNOTEGrahamKnuthPatashnik19899.67-24">[22]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=a}^{b-1}f(k)=\int _{a}^{b}f(x)\,dx+\sum _{k=1}^{m}{\frac {B_{k}^{-}}{k!}}(f^{(k-1)}(b)-f^{(k-1)}(a))+R_{-}(f,m).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msubsup> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=a}^{b-1}f(k)=\int _{a}^{b}f(x)\,dx+\sum _{k=1}^{m}{\frac {B_{k}^{-}}{k!}}(f^{(k-1)}(b)-f^{(k-1)}(a))+R_{-}(f,m).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d48aa0a79f6f893793e8fc023e9fd9535059456" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:67.97ex; height:7.343ex;" alt="{\displaystyle \sum _{k=a}^{b-1}f(k)=\int _{a}^{b}f(x)\,dx+\sum _{k=1}^{m}{\frac {B_{k}^{-}}{k!}}(f^{(k-1)}(b)-f^{(k-1)}(a))+R_{-}(f,m).}"></dd></dl> This formulation assumes the convention B− 1 = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠. Using the convention B+ 1 = +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ the formula becomes <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=a+1}^{b}f(k)=\int _{a}^{b}f(x)\,dx+\sum _{k=1}^{m}{\frac {B_{k}^{+}}{k!}}(f^{(k-1)}(b)-f^{(k-1)}(a))+R_{+}(f,m).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=a+1}^{b}f(k)=\int _{a}^{b}f(x)\,dx+\sum _{k=1}^{m}{\frac {B_{k}^{+}}{k!}}(f^{(k-1)}(b)-f^{(k-1)}(a))+R_{+}(f,m).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03e4ac61547e2d4db3477740b1a93b6f78619724" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:69.721ex; height:7.509ex;" alt="{\displaystyle \sum _{k=a+1}^{b}f(k)=\int _{a}^{b}f(x)\,dx+\sum _{k=1}^{m}{\frac {B_{k}^{+}}{k!}}(f^{(k-1)}(b)-f^{(k-1)}(a))+R_{+}(f,m).}"></dd></dl> Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(0)}=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{(0)}=f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c605cc879751dbc6e9e1f3d8921b36fa54993f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.031ex; height:3.176ex;" alt="{\displaystyle f^{(0)}=f}"> (i.e. the zeroth-order derivative of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is just <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">). Moreover, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(-1)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{(-1)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff92eaa7055a020ae8451ba6f52c729060846f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.933ex; height:3.176ex;" alt="{\displaystyle f^{(-1)}}"> denote an <a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">. By the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a>, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,dx=f^{(-1)}(b)-f^{(-1)}(a).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,dx=f^{(-1)}(b)-f^{(-1)}(a).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9756fef5266773bf4ec269ea4530b8d16e1c0139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.436ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx=f^{(-1)}(b)-f^{(-1)}(a).}"></dd></dl> Thus the last formula can be further simplified to the following succinct form of the Euler–Maclaurin formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=a+1}^{b}f(k)=\sum _{k=0}^{m}{\frac {B_{k}}{k!}}(f^{(k-1)}(b)-f^{(k-1)}(a))+R(f,m).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=a+1}^{b}f(k)=\sum _{k=0}^{m}{\frac {B_{k}}{k!}}(f^{(k-1)}(b)-f^{(k-1)}(a))+R(f,m).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/133b1a0390bd0875afdd27f3bd3aa4ef7da7a77e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.808ex; height:7.509ex;" alt="{\displaystyle \sum _{k=a+1}^{b}f(k)=\sum _{k=0}^{m}{\frac {B_{k}}{k!}}(f^{(k-1)}(b)-f^{(k-1)}(a))+R(f,m).}"></dd></dl> This form is for example the source for the important Euler–Maclaurin expansion of the zeta function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\zeta (s)&=\sum _{k=0}^{m}{\frac {B_{k}^{+}}{k!}}s^{\overline {k-1}}+R(s,m)\\&={\frac {B_{0}}{0!}}s^{\overline {-1}}+{\frac {B_{1}^{+}}{1!}}s^{\overline {0}}+{\frac {B_{2}}{2!}}s^{\overline {1}}+\cdots +R(s,m)\\&={\frac {1}{s-1}}+{\frac {1}{2}}+{\frac {1}{12}}s+\cdots +R(s,m).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msup> <mo>+</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>0</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mrow> <mn>1</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo accent="false">¯</mo> </mover> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>1</mn> <mo accent="false">¯</mo> </mover> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mi>s</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\zeta (s)&=\sum _{k=0}^{m}{\frac {B_{k}^{+}}{k!}}s^{\overline {k-1}}+R(s,m)\\&={\frac {B_{0}}{0!}}s^{\overline {-1}}+{\frac {B_{1}^{+}}{1!}}s^{\overline {0}}+{\frac {B_{2}}{2!}}s^{\overline {1}}+\cdots +R(s,m)\\&={\frac {1}{s-1}}+{\frac {1}{2}}+{\frac {1}{12}}s+\cdots +R(s,m).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b09cc59e4f5a9888c1290acf375459fe9aa9852a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.724ex; margin-bottom: -0.281ex; width:49.08ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}\zeta (s)&=\sum _{k=0}^{m}{\frac {B_{k}^{+}}{k!}}s^{\overline {k-1}}+R(s,m)\\&={\frac {B_{0}}{0!}}s^{\overline {-1}}+{\frac {B_{1}^{+}}{1!}}s^{\overline {0}}+{\frac {B_{2}}{2!}}s^{\overline {1}}+\cdots +R(s,m)\\&={\frac {1}{s-1}}+{\frac {1}{2}}+{\frac {1}{12}}s+\cdots +R(s,m).\end{aligned}}}"></dd></dl> Here sk denotes the <a href="/wiki/Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">rising factorial power</a>.<a href="#cite_note-FOOTNOTEGrahamKnuthPatashnik19892.44,_2.52-25">[23]</a> Bernoulli numbers are also frequently used in other kinds of <a href="/wiki/Asymptotic_expansion" title="Asymptotic expansion">asymptotic expansions</a>. The following example is the classical Poincaré-type asymptotic expansion of the <a href="/wiki/Digamma_function" title="Digamma function">digamma function</a> ψ. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (z)\sim \ln z-\sum _{k=1}^{\infty }{\frac {B_{k}^{+}}{kz^{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>∼</mo> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mrow> <mi>k</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (z)\sim \ln z-\sum _{k=1}^{\infty }{\frac {B_{k}^{+}}{kz^{k}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fb93ffc723958c329e1482c13e3e580df40468f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.733ex; height:7.176ex;" alt="{\displaystyle \psi (z)\sim \ln z-\sum _{k=1}^{\infty }{\frac {B_{k}^{+}}{kz^{k}}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Sum_of_powers">Sum of powers</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=14" title="Edit section: Sum of powers">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Faulhaber%27s_formula" title="Faulhaber's formula">Faulhaber's formula</a></div> Bernoulli numbers feature prominently in the <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed form</a> expression of the sum of the mth powers of the first n positive integers. For m, n ≥ 0 define <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{m}(n)=\sum _{k=1}^{n}k^{m}=1^{m}+2^{m}+\cdots +n^{m}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{m}(n)=\sum _{k=1}^{n}k^{m}=1^{m}+2^{m}+\cdots +n^{m}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ee2791d600b205822db64e78238f6883a58069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.765ex; height:6.843ex;" alt="{\displaystyle S_{m}(n)=\sum _{k=1}^{n}k^{m}=1^{m}+2^{m}+\cdots +n^{m}.}"></dd></dl> This expression can always be rewritten as a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> in n of degree m + 1. The <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> of these polynomials are related to the Bernoulli numbers by Bernoulli's formula: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{m}(n)={\frac {1}{m+1}}\sum _{k=0}^{m}{\binom {m+1}{k}}B_{k}^{+}n^{m+1-k}=m!\sum _{k=0}^{m}{\frac {B_{k}^{+}n^{m+1-k}}{k!(m+1-k)!}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mi>m</mi> <mo>!</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{m}(n)={\frac {1}{m+1}}\sum _{k=0}^{m}{\binom {m+1}{k}}B_{k}^{+}n^{m+1-k}=m!\sum _{k=0}^{m}{\frac {B_{k}^{+}n^{m+1-k}}{k!(m+1-k)!}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69cbea3365ef6f0625632f215211637cdd89da49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:66.262ex; height:7.343ex;" alt="{\displaystyle S_{m}(n)={\frac {1}{m+1}}\sum _{k=0}^{m}{\binom {m+1}{k}}B_{k}^{+}n^{m+1-k}=m!\sum _{k=0}^{m}{\frac {B_{k}^{+}n^{m+1-k}}{k!(m+1-k)!}},}"></dd></dl> where <big><big>(</big></big>m + 1 k<big><big>)</big></big> denotes the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a>. For example, taking m to be 1 gives the <a href="/wiki/Triangular_number" title="Triangular number">triangular numbers</a> 0, 1, 3, 6, ... <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A000217" class="extiw" title="oeis:A000217">A000217</a>. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+2+\cdots +n={\frac {1}{2}}(B_{0}n^{2}+2B_{1}^{+}n^{1})={\tfrac {1}{2}}(n^{2}+n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+2+\cdots +n={\frac {1}{2}}(B_{0}n^{2}+2B_{1}^{+}n^{1})={\tfrac {1}{2}}(n^{2}+n).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb92c53f31bd750634ea1ac86a6fcff5984801f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.761ex; height:5.176ex;" alt="{\displaystyle 1+2+\cdots +n={\frac {1}{2}}(B_{0}n^{2}+2B_{1}^{+}n^{1})={\tfrac {1}{2}}(n^{2}+n).}"></dd></dl> Taking m to be 2 gives the <a href="/wiki/Square_pyramidal_number" title="Square pyramidal number">square pyramidal numbers</a> 0, 1, 5, 14, ... <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A000330" class="extiw" title="oeis:A000330">A000330</a>. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{2}+2^{2}+\cdots +n^{2}={\frac {1}{3}}(B_{0}n^{3}+3B_{1}^{+}n^{2}+3B_{2}n^{1})={\tfrac {1}{3}}\left(n^{3}+{\tfrac {3}{2}}n^{2}+{\tfrac {1}{2}}n\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1^{2}+2^{2}+\cdots +n^{2}={\frac {1}{3}}(B_{0}n^{3}+3B_{1}^{+}n^{2}+3B_{2}n^{1})={\tfrac {1}{3}}\left(n^{3}+{\tfrac {3}{2}}n^{2}+{\tfrac {1}{2}}n\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2504da3eebf89de1c9bc0c65ac8f5a66ea81999f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:73.54ex; height:5.176ex;" alt="{\displaystyle 1^{2}+2^{2}+\cdots +n^{2}={\frac {1}{3}}(B_{0}n^{3}+3B_{1}^{+}n^{2}+3B_{2}n^{1})={\tfrac {1}{3}}\left(n^{3}+{\tfrac {3}{2}}n^{2}+{\tfrac {1}{2}}n\right).}"></dd></dl> Some authors use the alternate convention for Bernoulli numbers and state Bernoulli's formula in this way: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{m}(n)={\frac {1}{m+1}}\sum _{k=0}^{m}(-1)^{k}{\binom {m+1}{k}}B_{k}^{-{}}n^{m+1-k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> </msubsup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{m}(n)={\frac {1}{m+1}}\sum _{k=0}^{m}(-1)^{k}{\binom {m+1}{k}}B_{k}^{-{}}n^{m+1-k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f63f1948e2074fb95af79cf6e87b8a36bc6aa53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.584ex; height:7.009ex;" alt="{\displaystyle S_{m}(n)={\frac {1}{m+1}}\sum _{k=0}^{m}(-1)^{k}{\binom {m+1}{k}}B_{k}^{-{}}n^{m+1-k}.}"></dd></dl> Bernoulli's formula is sometimes called <a href="/wiki/Faulhaber%27s_formula" title="Faulhaber's formula">Faulhaber's formula</a> after <a href="/wiki/Johann_Faulhaber" title="Johann Faulhaber">Johann Faulhaber</a> who also found remarkable ways to calculate <a href="/wiki/Sums_of_powers" title="Sums of powers">sums of powers</a>. Faulhaber's formula was generalized by V. Guo and J. Zeng to a <a href="/wiki/Q-analog" title="Q-analog">q-analog</a>.<a href="#cite_note-GuoZeng2005-26">[24]</a> <div class="mw-heading mw-heading3"><h3 id="Taylor_series">Taylor series</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=15" title="Edit section: Taylor series">edit</a>]</div> The Bernoulli numbers appear in the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> expansion of many <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> and <a href="/wiki/Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic functions</a>. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\tan x&={\hphantom {1 \over x}}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}}\;x^{2n-1},&&\left|x\right|<{\frac {\pi }{2}}.\\\cot x&={1 \over x}\sum _{n=0}^{\infty }{\frac {(-1)^{n}B_{2n}(2x)^{2n}}{(2n)!}},&0<&|x|<\pi .\\\tanh x&={\hphantom {1 \over x}}\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}}\;x^{2n-1},&&|x|<{\frac {\pi }{2}}.\\\coth x&={1 \over x}\sum _{n=0}^{\infty }{\frac {B_{2n}(2x)^{2n}}{(2n)!}},&0<&|x|<\pi .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded height="0" depth="0"> <mphantom> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mphantom> </mpadded> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd /> <mtd> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mi>cot</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mn>0</mn> <mo><</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>π</mi> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mi>tanh</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded height="0" depth="0"> <mphantom> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mphantom> </mpadded> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mi>coth</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mn>0</mn> <mo><</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>π</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan x&={\hphantom {1 \over x}}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}}\;x^{2n-1},&&\left|x\right|<{\frac {\pi }{2}}.\\\cot x&={1 \over x}\sum _{n=0}^{\infty }{\frac {(-1)^{n}B_{2n}(2x)^{2n}}{(2n)!}},&0<&|x|<\pi .\\\tanh x&={\hphantom {1 \over x}}\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}}\;x^{2n-1},&&|x|<{\frac {\pi }{2}}.\\\coth x&={1 \over x}\sum _{n=0}^{\infty }{\frac {B_{2n}(2x)^{2n}}{(2n)!}},&0<&|x|<\pi .\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6348a0786a36c7298ed7237d01f8aaa65b08f4f3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.574ex; margin-bottom: -0.264ex; width:65.032ex; height:28.843ex;" alt="{\displaystyle {\begin{aligned}\tan x&={\hphantom {1 \over x}}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}}\;x^{2n-1},&&\left|x\right|<{\frac {\pi }{2}}.\\\cot x&={1 \over x}\sum _{n=0}^{\infty }{\frac {(-1)^{n}B_{2n}(2x)^{2n}}{(2n)!}},&0<&|x|<\pi .\\\tanh x&={\hphantom {1 \over x}}\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}}\;x^{2n-1},&&|x|<{\frac {\pi }{2}}.\\\coth x&={1 \over x}\sum _{n=0}^{\infty }{\frac {B_{2n}(2x)^{2n}}{(2n)!}},&0<&|x|<\pi .\end{aligned}}}"> <div class="mw-heading mw-heading3"><h3 id="Laurent_series">Laurent series</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=16" title="Edit section: Laurent series">edit</a>]</div> The Bernoulli numbers appear in the following <a href="/wiki/Laurent_series" title="Laurent series">Laurent series</a>:<a href="#cite_note-FOOTNOTEArfken1970463-27">[25]</a> <a href="/wiki/Digamma_function" title="Digamma function">Digamma function</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (z)=\ln z-\sum _{k=1}^{\infty }{\frac {B_{k}^{+{}}}{kz^{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> </msubsup> <mrow> <mi>k</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (z)=\ln z-\sum _{k=1}^{\infty }{\frac {B_{k}^{+{}}}{kz^{k}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07569a9bd6aea65575732976f724b412c8461dad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.733ex; height:7.176ex;" alt="{\displaystyle \psi (z)=\ln z-\sum _{k=1}^{\infty }{\frac {B_{k}^{+{}}}{kz^{k}}}}"> <div class="mw-heading mw-heading3"><h3 id="Use_in_topology">Use in topology</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=17" title="Edit section: Use in topology">edit</a>]</div> The <a href="/wiki/Kervaire%E2%80%93Milnor_formula" class="mw-redirect" title="Kervaire–Milnor formula">Kervaire–Milnor formula</a> for the order of the cyclic group of diffeomorphism classes of <a href="/wiki/Exotic_sphere" title="Exotic sphere">exotic (4n − 1)-spheres</a> which bound <a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">parallelizable manifolds</a> involves Bernoulli numbers. Let ESn be the number of such exotic spheres for n ≥ 2, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textit {ES}}_{n}=(2^{2n-2}-2^{4n-3})\operatorname {Numerator} \left({\frac {B_{4n}}{4n}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext class="MJX-tex-mathit" mathvariant="italic">ES</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>Numerator</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>n</mi> </mrow> </msub> <mrow> <mn>4</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textit {ES}}_{n}=(2^{2n-2}-2^{4n-3})\operatorname {Numerator} \left({\frac {B_{4n}}{4n}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7476d0c9db557917d57aa19fd7b8521de3d9ad36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.608ex; height:6.176ex;" alt="{\displaystyle {\textit {ES}}_{n}=(2^{2n-2}-2^{4n-3})\operatorname {Numerator} \left({\frac {B_{4n}}{4n}}\right).}"></dd></dl> The <a href="/wiki/Hirzebruch_signature_theorem#L_genus_and_the_Hirzebruch_signature_theorem" title="Hirzebruch signature theorem">Hirzebruch signature theorem</a> for the <a href="/wiki/Hirzebruch_signature_theorem#L_genus_and_the_Hirzebruch_signature_theorem" title="Hirzebruch signature theorem">L genus</a> of a <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth</a> <a href="/wiki/Orientability" title="Orientability">oriented</a> <a href="/wiki/Closed_manifold" title="Closed manifold">closed manifold</a> of <a href="/wiki/Dimension" title="Dimension">dimension</a> 4n also involves Bernoulli numbers. <div class="mw-heading mw-heading2"><h2 id="Connections_with_combinatorial_numbers">Connections with combinatorial numbers</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=18" title="Edit section: Connections with combinatorial numbers">edit</a>]</div> The connection of the Bernoulli number to various kinds of combinatorial numbers is based on the classical theory of finite differences and on the combinatorial interpretation of the Bernoulli numbers as an instance of a fundamental combinatorial principle, the <a href="/wiki/Inclusion%E2%80%93exclusion_principle" title="Inclusion–exclusion principle">inclusion–exclusion principle</a>. <div class="mw-heading mw-heading3"><h3 id="Connection_with_Worpitzky_numbers">Connection with Worpitzky numbers</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=19" title="Edit section: Connection with Worpitzky numbers">edit</a>]</div> The definition to proceed with was developed by Julius Worpitzky in 1883. Besides elementary arithmetic only the factorial function n! and the power function km is employed. The signless Worpitzky numbers are defined as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{n,k}=\sum _{v=0}^{k}(-1)^{v+k}(v+1)^{n}{\frac {k!}{v!(k-v)!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>+</mo> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>v</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>!</mo> </mrow> <mrow> <mi>v</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{n,k}=\sum _{v=0}^{k}(-1)^{v+k}(v+1)^{n}{\frac {k!}{v!(k-v)!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96e5c47ba703016d709a8f7e79edbacc324c28ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.175ex; height:7.343ex;" alt="{\displaystyle W_{n,k}=\sum _{v=0}^{k}(-1)^{v+k}(v+1)^{n}{\frac {k!}{v!(k-v)!}}.}"></dd></dl> They can also be expressed through the <a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling numbers of the second kind</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{n,k}=k!\left\{{n+1 \atop k+1}\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mo>!</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{n,k}=k!\left\{{n+1 \atop k+1}\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0e2bdfadc8ebe2b6548926352e878ebaa845fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.545ex; height:6.176ex;" alt="{\displaystyle W_{n,k}=k!\left\{{n+1 \atop k+1}\right\}.}"></dd></dl> A Bernoulli number is then introduced as an inclusion–exclusion sum of Worpitzky numbers weighted by the <a href="/wiki/Harmonic_progression_(mathematics)" title="Harmonic progression (mathematics)">harmonic sequence</a> 1, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠, ... <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}=\sum _{k=0}^{n}(-1)^{k}{\frac {W_{n,k}}{k+1}}\ =\ \sum _{k=0}^{n}{\frac {1}{k+1}}\sum _{v=0}^{k}(-1)^{v}(v+1)^{n}{k \choose v}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>v</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>v</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}=\sum _{k=0}^{n}(-1)^{k}{\frac {W_{n,k}}{k+1}}\ =\ \sum _{k=0}^{n}{\frac {1}{k+1}}\sum _{v=0}^{k}(-1)^{v}(v+1)^{n}{k \choose v}\ .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc0fd2fd113ca1aaa5e855d087f41304ac35c9ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.364ex; height:7.509ex;" alt="{\displaystyle B_{n}=\sum _{k=0}^{n}(-1)^{k}{\frac {W_{n,k}}{k+1}}\ =\ \sum _{k=0}^{n}{\frac {1}{k+1}}\sum _{v=0}^{k}(-1)^{v}(v+1)^{n}{k \choose v}\ .}"></dd></dl> <dl><dd>B0 = 1</dd> <dd>B1 = 1 − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠</dd> <dd>B2 = 1 − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/2⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/3⁠</dd> <dd>B3 = 1 − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠7/2⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠12/3⁠ − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠6/4⁠</dd> <dd>B4 = 1 − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠15/2⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠50/3⁠ − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠60/4⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠24/5⁠</dd> <dd>B5 = 1 − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠31/2⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠180/3⁠ − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠390/4⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠360/5⁠ − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠120/6⁠</dd> <dd>B6 = 1 − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠63/2⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠602/3⁠ − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2100/4⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3360/5⁠ − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2520/6⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠720/7⁠</dd></dl> This representation has B+ 1 = +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠. Consider the sequence sn, n ≥ 0. From Worpitzky's numbers <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A028246" class="extiw" title="oeis:A028246">A028246</a>, <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A163626" class="extiw" title="oeis:A163626">A163626</a> applied to s0, s0, s1, s0, s1, s2, s0, s1, s2, s3, ... is identical to the Akiyama–Tanigawa transform applied to sn (see <a href="#Connection_with_Stirling_numbers_of_the_first_kind">Connection with Stirling numbers of the first kind</a>). This can be seen via the table: <dl><dd><table style="text-align:center"> <caption>Identity of Worpitzky's representation and Akiyama–Tanigawa transform </caption> <tbody><tr> <td>1</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td>0</td> <td>1</td> <td></td> <td></td> <td></td> <td></td> <td>0</td> <td>0</td> <td>1</td> <td></td> <td></td> <td></td> <td>0</td> <td>0</td> <td>0</td> <td>1</td> <td></td> <td></td> <td>0</td> <td>0</td> <td>0</td> <td>0</td> <td>1</td> <td> </td></tr> <tr> <td>1</td> <td>−1</td> <td></td> <td></td> <td></td> <td></td> <td>0</td> <td>2</td> <td>−2</td> <td></td> <td></td> <td></td> <td>0</td> <td>0</td> <td>3</td> <td>−3</td> <td></td> <td></td> <td>0</td> <td>0</td> <td>0</td> <td>4</td> <td>−4</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <td>1</td> <td>−3</td> <td>2</td> <td></td> <td></td> <td></td> <td>0</td> <td>4</td> <td>−10</td> <td>6</td> <td></td> <td></td> <td>0</td> <td>0</td> <td>9</td> <td>−21</td> <td>12</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <td>1</td> <td>−7</td> <td>12</td> <td>−6</td> <td></td> <td></td> <td>0</td> <td>8</td> <td>−38</td> <td>54</td> <td>−24</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <td>1</td> <td>−15</td> <td>50</td> <td>−60</td> <td>24</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr> </tbody></table></dd></dl> The first row represents s0, s1, s2, s3, s4. Hence for the second fractional Euler numbers <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A198631" class="extiw" title="oeis:A198631">A198631</a> (n) / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A006519" class="extiw" title="oeis:A006519">A006519</a> (n + 1): <dl><dd>E0 = 1</dd> <dd>E1 = 1 − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠</dd> <dd>E2 = 1 − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/2⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/4⁠</dd> <dd>E3 = 1 − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠7/2⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠12/4⁠ − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠6/8⁠</dd> <dd>E4 = 1 − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠15/2⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠50/4⁠ − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠60/8⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠24/16⁠</dd> <dd>E5 = 1 − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠31/2⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠180/4⁠ − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠390/8⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠360/16⁠ − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠120/32⁠</dd> <dd>E6 = 1 − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠63/2⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠602/4⁠ − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2100/8⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3360/16⁠ − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2520/32⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠720/64⁠</dd></dl> A second formula representing the Bernoulli numbers by the Worpitzky numbers is for n ≥ 1 <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}={\frac {n}{2^{n+1}-2}}\sum _{k=0}^{n-1}(-2)^{-k}\,W_{n-1,k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}={\frac {n}{2^{n+1}-2}}\sum _{k=0}^{n-1}(-2)^{-k}\,W_{n-1,k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e104e02bbb0ade13e067504cfc83f888f32cc30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.151ex; height:7.509ex;" alt="{\displaystyle B_{n}={\frac {n}{2^{n+1}-2}}\sum _{k=0}^{n-1}(-2)^{-k}\,W_{n-1,k}.}"></dd></dl> The simplified second Worpitzky's representation of the second Bernoulli numbers is: <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A164555" class="extiw" title="oeis:A164555">A164555</a> (n + 1) / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A027642" class="extiw" title="oeis:A027642">A027642</a>(n + 1) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠n + 1/2n + 2 − 2⁠ × <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A198631" class="extiw" title="oeis:A198631">A198631</a>(n) / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A006519" class="extiw" title="oeis:A006519">A006519</a>(n + 1) which links the second Bernoulli numbers to the second fractional Euler numbers. The beginning is: <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠, 0, −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/30⁠, 0, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/42⁠, ... = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/14⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/15⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/62⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/21⁠, ...) × (1, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, 0, −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠, 0, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, ...)</dd></dl> The numerators of the first parentheses are <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A111701" class="extiw" title="oeis:A111701">A111701</a> (see <a href="#Connection_with_Stirling_numbers_of_the_first_kind">Connection with Stirling numbers of the first kind</a>). <div class="mw-heading mw-heading3"><h3 id="Connection_with_Stirling_numbers_of_the_second_kind">Connection with Stirling numbers of the second kind</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=20" title="Edit section: Connection with Stirling numbers of the second kind">edit</a>]</div> If one defines the <a href="/wiki/Bernoulli_polynomials" title="Bernoulli polynomials">Bernoulli polynomials</a> Bk(j) as:<a href="#cite_note-Rademacher1973-28">[26]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{k}(j)=k\sum _{m=0}^{k-1}{\binom {j}{m+1}}S(k-1,m)m!+B_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>j</mi> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mi>S</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mi>m</mi> <mo>!</mo> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{k}(j)=k\sum _{m=0}^{k-1}{\binom {j}{m+1}}S(k-1,m)m!+B_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b2e49e0cce9589d0112fc24b353452e4f0e7aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.689ex; height:7.343ex;" alt="{\displaystyle B_{k}(j)=k\sum _{m=0}^{k-1}{\binom {j}{m+1}}S(k-1,m)m!+B_{k}}"></dd></dl> where Bk for k = 0, 1, 2,... are the Bernoulli numbers, and S(k,m) is a <a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling number of the second kind</a>. One also has the following for Bernoulli polynomials,<a href="#cite_note-Rademacher1973-28">[26]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{k}(j)=\sum _{n=0}^{k}{\binom {k}{n}}B_{n}j^{k-n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{k}(j)=\sum _{n=0}^{k}{\binom {k}{n}}B_{n}j^{k-n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c606a1ede5c1c4edc71affcfdca9ec68d8bbb51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.217ex; height:7.343ex;" alt="{\displaystyle B_{k}(j)=\sum _{n=0}^{k}{\binom {k}{n}}B_{n}j^{k-n}.}"></dd></dl> The coefficient of j in <big><big>(</big></big>j m + 1<big><big>)</big></big> is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠(−1)m/m + 1⁠. Comparing the coefficient of j in the two expressions of Bernoulli polynomials, one has: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{k}=\sum _{m=0}^{k-1}(-1)^{m}{\frac {m!}{m+1}}S(k-1,m)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo>!</mo> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mi>S</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{k}=\sum _{m=0}^{k-1}(-1)^{m}{\frac {m!}{m+1}}S(k-1,m)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5e433dad6c9016f4066d3b68b1302dd3e1a5926" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.426ex; height:7.343ex;" alt="{\displaystyle B_{k}=\sum _{m=0}^{k-1}(-1)^{m}{\frac {m!}{m+1}}S(k-1,m)}"></dd></dl> (resulting in B1 = +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠) which is an explicit formula for Bernoulli numbers and can be used to prove <a href="/wiki/Von_Staudt%E2%80%93Clausen_theorem" title="Von Staudt–Clausen theorem">Von-Staudt Clausen theorem</a>.<a href="#cite_note-Boole1880-29">[27]</a><a href="#cite_note-Gould1972-30">[28]</a><a href="#cite_note-Apostol2010_197-31">[29]</a> <div class="mw-heading mw-heading3"><h3 id="Connection_with_Stirling_numbers_of_the_first_kind">Connection with Stirling numbers of the first kind</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=21" title="Edit section: Connection with Stirling numbers of the first kind">edit</a>]</div> The two main formulas relating the unsigned <a href="/wiki/Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling numbers of the first kind</a> <big><big>[</big></big>n m<big><big>]</big></big> to the Bernoulli numbers (with B1 = +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠) are <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{m!}}\sum _{k=0}^{m}(-1)^{k}\left[{m+1 \atop k+1}\right]B_{k}={\frac {1}{m+1}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>m</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{m!}}\sum _{k=0}^{m}(-1)^{k}\left[{m+1 \atop k+1}\right]B_{k}={\frac {1}{m+1}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b5b65309bc0ec139514174501d910ac794fff06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.441ex; height:7.009ex;" alt="{\displaystyle {\frac {1}{m!}}\sum _{k=0}^{m}(-1)^{k}\left[{m+1 \atop k+1}\right]B_{k}={\frac {1}{m+1}},}"></dd></dl> and the inversion of this sum (for n ≥ 0, m ≥ 0) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{m!}}\sum _{k=0}^{m}(-1)^{k}\left[{m+1 \atop k+1}\right]B_{n+k}=A_{n,m}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>m</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{m!}}\sum _{k=0}^{m}(-1)^{k}\left[{m+1 \atop k+1}\right]B_{n+k}=A_{n,m}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35e17423d768b3f1b45db0d78075b6b4911ea350" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.688ex; height:7.009ex;" alt="{\displaystyle {\frac {1}{m!}}\sum _{k=0}^{m}(-1)^{k}\left[{m+1 \atop k+1}\right]B_{n+k}=A_{n,m}.}"></dd></dl> Here the number An,m are the rational Akiyama–Tanigawa numbers, the first few of which are displayed in the following table. <dl><dd><table class="wikitable" style="text-align:center"> <caption>Akiyama–Tanigawa number </caption> <tbody><tr> <th style="background:#EAECF0;background:linear-gradient(to top right,#EAECF0 49%,#AAA 49.5%,#AAA 50.5%,#EAECF0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">m</div><div style="margin-right:2em;text-align:left">n</div></th> <th>0</th> <th>1</th> <th>2</th> <th>3</th> <th>4 </th></tr> <tr> <th>0 </th> <td>1</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/5⁠ </td></tr> <tr> <th>1 </th> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/5⁠</td> <td>... </td></tr> <tr> <th>2 </th> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/20⁠</td> <td>...</td> <td>... </td></tr> <tr> <th>3 </th> <td>0</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/30⁠</td> <td>...</td> <td>...</td> <td>... </td></tr> <tr> <th>4 </th> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/30⁠</td> <td>...</td> <td>...</td> <td>...</td> <td>... </td></tr></tbody></table></dd></dl> The Akiyama–Tanigawa numbers satisfy a simple recurrence relation which can be exploited to iteratively compute the Bernoulli numbers. This leads to the algorithm shown in the section 'algorithmic description' above. See <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A051714" class="extiw" title="oeis:A051714">A051714</a>/<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A051715" class="extiw" title="oeis:A051715">A051715</a>. An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. If the main diagonal is zeroes = <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A000004" class="extiw" title="oeis:A000004">A000004</a>, the autosequence is of the first kind. Example: <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A000045" class="extiw" title="oeis:A000045">A000045</a>, the Fibonacci numbers. If the main diagonal is the first upper diagonal multiplied by 2, it is of the second kind. Example: <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A164555" class="extiw" title="oeis:A164555">A164555</a>/<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A027642" class="extiw" title="oeis:A027642">A027642</a>, the second Bernoulli numbers (see <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A190339" class="extiw" title="oeis:A190339">A190339</a>). The Akiyama–Tanigawa transform applied to 2−n = 1/<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A000079" class="extiw" title="oeis:A000079">A000079</a> leads to <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A198631" class="extiw" title="oeis:A198631">A198631</a> (n) / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A06519" class="extiw" title="oeis:A06519">A06519</a> (n + 1). Hence: <dl><dd><table class="wikitable" style="text-align:center"> <caption>Akiyama–Tanigawa transform for the second Euler numbers </caption> <tbody><tr> <th style="background:#EAECF0;background:linear-gradient(to top right,#EAECF0 49%,#AAA 49.5%,#AAA 50.5%,#EAECF0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">m</div><div style="margin-right:2em;text-align:left">n</div></th> <th>0</th> <th>1</th> <th>2</th> <th>3</th> <th>4 </th></tr> <tr> <th>0 </th> <td>1</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/8⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/16⁠ </td></tr> <tr> <th>1 </th> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/8⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td>... </td></tr> <tr> <th>2 </th> <td>0</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/8⁠</td> <td>...</td> <td>... </td></tr> <tr> <th>3 </th> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td>...</td> <td>...</td> <td>... </td></tr> <tr> <th>4 </th> <td>0</td> <td>...</td> <td>...</td> <td>...</td> <td>... </td></tr></tbody></table></dd></dl> See <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A209308" class="extiw" title="oeis:A209308">A209308</a> and <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A227577" class="extiw" title="oeis:A227577">A227577</a>. <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A198631" class="extiw" title="oeis:A198631">A198631</a> (n) / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A006519" class="extiw" title="oeis:A006519">A006519</a> (n + 1) are the second (fractional) Euler numbers and an autosequence of the second kind. <dl><dd>(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A164555" class="extiw" title="oeis:A164555">A164555</a> (n + 2)/<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A027642" class="extiw" title="oeis:A027642">A027642</a> (n + 2)⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠, 0, −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/30⁠, 0, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/42⁠, ...) × ( <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2n + 3 − 2/n + 2⁠ = 3, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠14/3⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠15/2⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠62/5⁠, 21, ...) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A198631" class="extiw" title="oeis:A198631">A198631</a> (n + 1)/<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A006519" class="extiw" title="oeis:A006519">A006519</a> (n + 2)⁠ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, 0, −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠, 0, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, ....</dd></dl> Also valuable for <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A027641" class="extiw" title="oeis:A027641">A027641</a> / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A027642" class="extiw" title="oeis:A027642">A027642</a> (see <a href="#Connection_with_Worpitzky_numbers">Connection with Worpitzky numbers</a>). <div class="mw-heading mw-heading3"><h3 id="Connection_with_Pascal's_triangle">Connection with Pascal's triangle</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=22" title="Edit section: Connection with Pascal's triangle">edit</a>]</div> There are formulas connecting Pascal's triangle to Bernoulli numbers<a href="#cite_note-32">[c]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}^{+}={\frac {|A_{n}|}{(n+1)!}}~~~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}^{+}={\frac {|A_{n}|}{(n+1)!}}~~~}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01a8a40cfe1e2b5738e2d3714bf9f5ccd589b1af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.805ex; height:6.509ex;" alt="{\displaystyle B_{n}^{+}={\frac {|A_{n}|}{(n+1)!}}~~~}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |A_{n}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |A_{n}|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4dd3a83e56c3340a7f5e1f3edb8adf6128f7e11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle |A_{n}|}"> is the determinant of a n-by-n <a href="/wiki/Hessenberg_matrix" title="Hessenberg matrix">Hessenberg matrix</a> part of <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a> whose elements are: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i,k}={\begin{cases}0&{\text{if }}k>1+i\\{i+1 \choose k-1}&{\text{otherwise}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>k</mi> <mo>></mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i,k}={\begin{cases}0&{\text{if }}k>1+i\\{i+1 \choose k-1}&{\text{otherwise}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4f7b80a1d52102e1d9205d944f21f3ad5ece1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.4ex; height:6.509ex;" alt="{\displaystyle a_{i,k}={\begin{cases}0&{\text{if }}k>1+i\\{i+1 \choose k-1}&{\text{otherwise}}\end{cases}}}"> Example: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{6}^{+}={\frac {\det {\begin{pmatrix}1&2&0&0&0&0\\1&3&3&0&0&0\\1&4&6&4&0&0\\1&5&10&10&5&0\\1&6&15&20&15&6\\1&7&21&35&35&21\end{pmatrix}}}{7!}}={\frac {120}{5040}}={\frac {1}{42}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mn>10</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>15</mn> </mtd> <mtd> <mn>20</mn> </mtd> <mtd> <mn>15</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>21</mn> </mtd> <mtd> <mn>35</mn> </mtd> <mtd> <mn>35</mn> </mtd> <mtd> <mn>21</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>120</mn> <mn>5040</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>42</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{6}^{+}={\frac {\det {\begin{pmatrix}1&2&0&0&0&0\\1&3&3&0&0&0\\1&4&6&4&0&0\\1&5&10&10&5&0\\1&6&15&20&15&6\\1&7&21&35&35&21\end{pmatrix}}}{7!}}={\frac {120}{5040}}={\frac {1}{42}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c58edf4420bbc19cfa9a53e831482f203384f6a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:53.726ex; height:22.009ex;" alt="{\displaystyle B_{6}^{+}={\frac {\det {\begin{pmatrix}1&2&0&0&0&0\\1&3&3&0&0&0\\1&4&6&4&0&0\\1&5&10&10&5&0\\1&6&15&20&15&6\\1&7&21&35&35&21\end{pmatrix}}}{7!}}={\frac {120}{5040}}={\frac {1}{42}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Connection_with_Eulerian_numbers">Connection with Eulerian numbers</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=23" title="Edit section: Connection with Eulerian numbers">edit</a>]</div> There are formulas connecting <a href="/wiki/Eulerian_number" title="Eulerian number">Eulerian numbers</a> <big><big>⟨</big></big>n m<big><big>⟩</big></big> to Bernoulli numbers: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sum _{m=0}^{n}(-1)^{m}\left\langle {n \atop m}\right\rangle &=2^{n+1}(2^{n+1}-1){\frac {B_{n+1}}{n+1}},\\\sum _{m=0}^{n}(-1)^{m}\left\langle {n \atop m}\right\rangle {\binom {n}{m}}^{-1}&=(n+1)B_{n}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> </mrow> <mo>⟩</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> </mrow> <mo>⟩</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{m=0}^{n}(-1)^{m}\left\langle {n \atop m}\right\rangle &=2^{n+1}(2^{n+1}-1){\frac {B_{n+1}}{n+1}},\\\sum _{m=0}^{n}(-1)^{m}\left\langle {n \atop m}\right\rangle {\binom {n}{m}}^{-1}&=(n+1)B_{n}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8694e8314b6ce6cb20dccb08475917493a1f956d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:49.511ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\sum _{m=0}^{n}(-1)^{m}\left\langle {n \atop m}\right\rangle &=2^{n+1}(2^{n+1}-1){\frac {B_{n+1}}{n+1}},\\\sum _{m=0}^{n}(-1)^{m}\left\langle {n \atop m}\right\rangle {\binom {n}{m}}^{-1}&=(n+1)B_{n}.\end{aligned}}}"></dd></dl> Both formulae are valid for n ≥ 0 if B1 is set to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠. If B1 is set to −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ they are valid only for n ≥ 1 and n ≥ 2 respectively. <div class="mw-heading mw-heading2"><h2 id="A_binary_tree_representation">A binary tree representation</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=24" title="Edit section: A binary tree representation">edit</a>]</div> The Stirling polynomials σn(x) are related to the Bernoulli numbers by Bn = n!σn(1). S. C. Woon described an algorithm to compute σn(1) as a binary tree:<a href="#cite_note-Woon1997-33">[30]</a> <dl><dd><a href="/wiki/File:SCWoonTree.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/1a/SCWoonTree.png" decoding="async" width="540" height="240" class="mw-file-element" data-file-width="540" data-file-height="240" /></a></dd></dl> Woon's recursive algorithm (for n ≥ 1) starts by assigning to the root node N = [1,2]. Given a node N = [a1, a2, ..., ak] of the tree, the left child of the node is L(N) = [−a1, a2 + 1, a3, ..., ak] and the right child R(N) = [a1, 2, a2, ..., ak]. A node N = [a1, a2, ..., ak] is written as ±[a2, ..., ak] in the initial part of the tree represented above with ± denoting the sign of a1. Given a node N the factorial of N is defined as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N!=a_{1}\prod _{k=2}^{\operatorname {length} (N)}a_{k}!.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>!</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>length</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>!</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N!=a_{1}\prod _{k=2}^{\operatorname {length} (N)}a_{k}!.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42bde0ffc3e10f639e8582f4fab7e625a78eea88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.695ex; height:7.676ex;" alt="{\displaystyle N!=a_{1}\prod _{k=2}^{\operatorname {length} (N)}a_{k}!.}"></dd></dl> Restricted to the nodes N of a fixed tree-level n the sum of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/N!⁠ is σn(1), thus <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}=\sum _{\stackrel {N{\text{ node of}}}{{\text{ tree-level }}n}}{\frac {n!}{N!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mtext> tree-level </mtext> </mrow> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> node of</mtext> </mrow> </mrow> </mover> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mi>N</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}=\sum _{\stackrel {N{\text{ node of}}}{{\text{ tree-level }}n}}{\frac {n!}{N!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d075d31c7dbec3ee3c99ba61076f850747b5878a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:19.344ex; height:8.009ex;" alt="{\displaystyle B_{n}=\sum _{\stackrel {N{\text{ node of}}}{{\text{ tree-level }}n}}{\frac {n!}{N!}}.}"></dd></dl> For example: <dl><dd>B1 = 1!(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2!⁠)</dd> <dd>B2 = 2!(−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3!⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2!2!⁠)</dd> <dd>B3 = 3!(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4!⁠ − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2!3!⁠ − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3!2!⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2!2!2!⁠)</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Integral_representation_and_continuation">Integral representation and continuation</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=25" title="Edit section: Integral representation and continuation">edit</a>]</div> The <a href="/wiki/Integral" title="Integral">integral</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b(s)=2e^{si\pi /2}\int _{0}^{\infty }{\frac {st^{s}}{1-e^{2\pi t}}}{\frac {dt}{t}}={\frac {s!}{2^{s-1}}}{\frac {\zeta (s)}{{}\pi ^{s}{}}}(-i)^{s}={\frac {2s!\zeta (s)}{(2\pi i)^{s}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>i</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>t</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo>!</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>s</mi> <mo>!</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b(s)=2e^{si\pi /2}\int _{0}^{\infty }{\frac {st^{s}}{1-e^{2\pi t}}}{\frac {dt}{t}}={\frac {s!}{2^{s-1}}}{\frac {\zeta (s)}{{}\pi ^{s}{}}}(-i)^{s}={\frac {2s!\zeta (s)}{(2\pi i)^{s}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e98f535a5131810e6989492d7735cf1db42b76bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.2ex; height:6.509ex;" alt="{\displaystyle b(s)=2e^{si\pi /2}\int _{0}^{\infty }{\frac {st^{s}}{1-e^{2\pi t}}}{\frac {dt}{t}}={\frac {s!}{2^{s-1}}}{\frac {\zeta (s)}{{}\pi ^{s}{}}}(-i)^{s}={\frac {2s!\zeta (s)}{(2\pi i)^{s}}}}"></dd></dl> has as special values b(2n) = B2n for n > 0. For example, b(3) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/2⁠ζ(3)π−3i and b(5) = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠15/2⁠ζ(5)π−5i. Here, ζ is the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>, and i is the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>. Leonhard Euler (Opera Omnia, Ser. 1, Vol. 10, p. 351) considered these numbers and calculated <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p&={\frac {3}{2\pi ^{3}}}\left(1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+\cdots \right)=0.0581522\ldots \\q&={\frac {15}{2\pi ^{5}}}\left(1+{\frac {1}{2^{5}}}+{\frac {1}{3^{5}}}+\cdots \right)=0.0254132\ldots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>p</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.0581522</mn> <mo>…</mo> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>15</mn> <mrow> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.0254132</mn> <mo>…</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p&={\frac {3}{2\pi ^{3}}}\left(1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+\cdots \right)=0.0581522\ldots \\q&={\frac {15}{2\pi ^{5}}}\left(1+{\frac {1}{2^{5}}}+{\frac {1}{3^{5}}}+\cdots \right)=0.0254132\ldots \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22810589a0b9c0b4a99435f8d479cd865a9fe43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:47.882ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}p&={\frac {3}{2\pi ^{3}}}\left(1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+\cdots \right)=0.0581522\ldots \\q&={\frac {15}{2\pi ^{5}}}\left(1+{\frac {1}{2^{5}}}+{\frac {1}{3^{5}}}+\cdots \right)=0.0254132\ldots \end{aligned}}}"></dd></dl> Another similar <a href="/wiki/Integral" title="Integral">integral</a> representation is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b(s)=-{\frac {e^{si\pi /2}}{2^{s}-1}}\int _{0}^{\infty }{\frac {st^{s}}{\sinh \pi t}}{\frac {dt}{t}}={\frac {2e^{si\pi /2}}{2^{s}-1}}\int _{0}^{\infty }{\frac {e^{\pi t}st^{s}}{1-e^{2\pi t}}}{\frac {dt}{t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>i</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mrow> <mrow> <mi>sinh</mi> <mo>⁡</mo> <mi>π</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>i</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mi>t</mi> </mrow> </msup> <mi>s</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>t</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <mi>t</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b(s)=-{\frac {e^{si\pi /2}}{2^{s}-1}}\int _{0}^{\infty }{\frac {st^{s}}{\sinh \pi t}}{\frac {dt}{t}}={\frac {2e^{si\pi /2}}{2^{s}-1}}\int _{0}^{\infty }{\frac {e^{\pi t}st^{s}}{1-e^{2\pi t}}}{\frac {dt}{t}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa8c62eac806c152513a145151532f8c29809379" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:58.859ex; height:6.176ex;" alt="{\displaystyle b(s)=-{\frac {e^{si\pi /2}}{2^{s}-1}}\int _{0}^{\infty }{\frac {st^{s}}{\sinh \pi t}}{\frac {dt}{t}}={\frac {2e^{si\pi /2}}{2^{s}-1}}\int _{0}^{\infty }{\frac {e^{\pi t}st^{s}}{1-e^{2\pi t}}}{\frac {dt}{t}}.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="The_relation_to_the_Euler_numbers_and_π">The relation to the Euler numbers and π</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=26" title="Edit section: The relation to the Euler numbers and π">edit</a>]</div> The <a href="/wiki/Euler_number" class="mw-redirect" title="Euler number">Euler numbers</a> are a sequence of integers intimately connected with the Bernoulli numbers. Comparing the asymptotic expansions of the Bernoulli and the Euler numbers shows that the Euler numbers E2n are in magnitude approximately <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/π⁠(42n − 22n) times larger than the Bernoulli numbers B2n. In consequence: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \sim 2(2^{2n}-4^{2n}){\frac {B_{2n}}{E_{2n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>∼</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \sim 2(2^{2n}-4^{2n}){\frac {B_{2n}}{E_{2n}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9cbfc31f17c4896feeb94e57d98d7cd5fab3c3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.936ex; height:5.509ex;" alt="{\displaystyle \pi \sim 2(2^{2n}-4^{2n}){\frac {B_{2n}}{E_{2n}}}.}"></dd></dl> This asymptotic equation reveals that π lies in the common root of both the Bernoulli and the Euler numbers. In fact π could be computed from these rational approximations. Bernoulli numbers can be expressed through the Euler numbers and vice versa. Since, for odd n, Bn = En = 0 (with the exception B1), it suffices to consider the case when n is even. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}B_{n}&=\sum _{k=0}^{n-1}{\binom {n-1}{k}}{\frac {n}{4^{n}-2^{n}}}E_{k}&n&=2,4,6,\ldots \\[6pt]E_{n}&=\sum _{k=1}^{n}{\binom {n}{k-1}}{\frac {2^{k}-4^{k}}{k}}B_{k}&n&=2,4,6,\ldots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi>n</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mo>…</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>−</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mi>k</mi> </mfrac> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mi>n</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mo>…</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}B_{n}&=\sum _{k=0}^{n-1}{\binom {n-1}{k}}{\frac {n}{4^{n}-2^{n}}}E_{k}&n&=2,4,6,\ldots \\[6pt]E_{n}&=\sum _{k=1}^{n}{\binom {n}{k-1}}{\frac {2^{k}-4^{k}}{k}}B_{k}&n&=2,4,6,\ldots \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2745968e7bb17361e79ceeae902aae954ef5bf16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:49.086ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}B_{n}&=\sum _{k=0}^{n-1}{\binom {n-1}{k}}{\frac {n}{4^{n}-2^{n}}}E_{k}&n&=2,4,6,\ldots \\[6pt]E_{n}&=\sum _{k=1}^{n}{\binom {n}{k-1}}{\frac {2^{k}-4^{k}}{k}}B_{k}&n&=2,4,6,\ldots \end{aligned}}}"></dd></dl> These conversion formulas express a connection between the Bernoulli and the Euler numbers. But more important, there is a deep arithmetic root common to both kinds of numbers, which can be expressed through a more fundamental sequence of numbers, also closely tied to π. These numbers are defined for n ≥ 1 as<a href="#cite_note-34">[31]</a><a href="#cite_note-Elkies2003-35">[32]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}=2\left({\frac {2}{\pi }}\right)^{n}\sum _{k=0}^{\infty }{\frac {(-1)^{kn}}{(2k+1)^{n}}}=2\left({\frac {2}{\pi }}\right)^{n}\lim _{K\to \infty }\sum _{k=-K}^{K}(4k+1)^{-n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi>K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>4</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}=2\left({\frac {2}{\pi }}\right)^{n}\sum _{k=0}^{\infty }{\frac {(-1)^{kn}}{(2k+1)^{n}}}=2\left({\frac {2}{\pi }}\right)^{n}\lim _{K\to \infty }\sum _{k=-K}^{K}(4k+1)^{-n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e46779b4796bbb3df4e815d38e2360085d169f13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:60.876ex; height:7.509ex;" alt="{\displaystyle S_{n}=2\left({\frac {2}{\pi }}\right)^{n}\sum _{k=0}^{\infty }{\frac {(-1)^{kn}}{(2k+1)^{n}}}=2\left({\frac {2}{\pi }}\right)^{n}\lim _{K\to \infty }\sum _{k=-K}^{K}(4k+1)^{-n}.}"></dd></dl> The magic of these numbers lies in the fact that they turn out to be rational numbers. This was first proved by <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> in a landmark paper De summis serierum reciprocarum (On the sums of series of reciprocals) and has fascinated mathematicians ever since.<a href="#cite_note-Euler1735-36">[33]</a> The first few of these numbers are <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}=1,1,{\frac {1}{2}},{\frac {1}{3}},{\frac {5}{24}},{\frac {2}{15}},{\frac {61}{720}},{\frac {17}{315}},{\frac {277}{8064}},{\frac {62}{2835}},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>24</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>15</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>61</mn> <mn>720</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>17</mn> <mn>315</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>277</mn> <mn>8064</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>62</mn> <mn>2835</mn> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}=1,1,{\frac {1}{2}},{\frac {1}{3}},{\frac {5}{24}},{\frac {2}{15}},{\frac {61}{720}},{\frac {17}{315}},{\frac {277}{8064}},{\frac {62}{2835}},\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19fa9de556d7ec245e434471643ace171b155ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:51.068ex; height:5.343ex;" alt="{\displaystyle S_{n}=1,1,{\frac {1}{2}},{\frac {1}{3}},{\frac {5}{24}},{\frac {2}{15}},{\frac {61}{720}},{\frac {17}{315}},{\frac {277}{8064}},{\frac {62}{2835}},\ldots }"> (<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A099612" class="extiw" title="oeis:A099612">A099612</a> / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A099617" class="extiw" title="oeis:A099617">A099617</a>)</dd></dl> These are the coefficients in the expansion of sec x + tan x. The Bernoulli numbers and Euler numbers can be understood as special views of these numbers, selected from the sequence Sn and scaled for use in special applications. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}B_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }[n{\text{ even}}]{\frac {n!}{2^{n}-4^{n}}}\,S_{n}\ ,&n&=2,3,\ldots \\E_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }[n{\text{ even}}]n!\,S_{n+1}&n&=0,1,\ldots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mrow> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> even</mtext> </mrow> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mtext> </mtext> <mo>,</mo> </mtd> <mtd> <mi>n</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mrow> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> even</mtext> </mrow> <mo stretchy="false">]</mo> <mi>n</mi> <mo>!</mo> <mspace width="thinmathspace" /> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi>n</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}B_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }[n{\text{ even}}]{\frac {n!}{2^{n}-4^{n}}}\,S_{n}\ ,&n&=2,3,\ldots \\E_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }[n{\text{ even}}]n!\,S_{n+1}&n&=0,1,\ldots \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/474673fce864f722449772fce7485f032fb41c6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:52.022ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}B_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }[n{\text{ even}}]{\frac {n!}{2^{n}-4^{n}}}\,S_{n}\ ,&n&=2,3,\ldots \\E_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }[n{\text{ even}}]n!\,S_{n+1}&n&=0,1,\ldots \end{aligned}}}"></dd></dl> The expression [n even] has the value 1 if n is even and 0 otherwise (<a href="/wiki/Iverson_bracket" title="Iverson bracket">Iverson bracket</a>). These identities show that the quotient of Bernoulli and Euler numbers at the beginning of this section is just the special case of Rn = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2Sn/Sn + 1⁠ when n is even. The Rn are rational approximations to π and two successive terms always enclose the true value of π. Beginning with n = 1 the sequence starts (<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A132049" class="extiw" title="oeis:A132049">A132049</a> / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A132050" class="extiw" title="oeis:A132050">A132050</a>): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2,4,3,{\frac {16}{5}},{\frac {25}{8}},{\frac {192}{61}},{\frac {427}{136}},{\frac {4352}{1385}},{\frac {12465}{3968}},{\frac {158720}{50521}},\ldots \quad \longrightarrow \pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>16</mn> <mn>5</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>25</mn> <mn>8</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>192</mn> <mn>61</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>427</mn> <mn>136</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4352</mn> <mn>1385</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>12465</mn> <mn>3968</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>158720</mn> <mn>50521</mn> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mspace width="1em" /> <mo stretchy="false">⟶</mo> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2,4,3,{\frac {16}{5}},{\frac {25}{8}},{\frac {192}{61}},{\frac {427}{136}},{\frac {4352}{1385}},{\frac {12465}{3968}},{\frac {158720}{50521}},\ldots \quad \longrightarrow \pi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/080346b24cdfb7a9653ca5174db0eacda80296b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:60.862ex; height:5.176ex;" alt="{\displaystyle 2,4,3,{\frac {16}{5}},{\frac {25}{8}},{\frac {192}{61}},{\frac {427}{136}},{\frac {4352}{1385}},{\frac {12465}{3968}},{\frac {158720}{50521}},\ldots \quad \longrightarrow \pi .}"></dd></dl> These rational numbers also appear in the last paragraph of Euler's paper cited above. Consider the Akiyama–Tanigawa transform for the sequence <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A046978" class="extiw" title="oeis:A046978">A046978</a> (n + 2) / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A016116" class="extiw" title="oeis:A016116">A016116</a> (n + 1): <dl><dd><table class="wikitable" style="text-align:right;"> <tbody><tr> <th>0 </th> <td>1</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠</td> <td>0</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/8⁠</td> <td>0 </td></tr> <tr> <th>1 </th> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠</td> <td>1</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/4⁠</td> <td>0</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/8⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/4⁠</td> <td> </td></tr> <tr> <th>2 </th> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠9/4⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/2⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/8⁠</td> <td></td> <td> </td></tr> <tr> <th>3 </th> <td>−1</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠7/2⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/4⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠15/2⁠</td> <td></td> <td></td> <td> </td></tr> <tr> <th>4 </th> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/2⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠11/2⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠99/4⁠</td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <th>5 </th> <td>8</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠77/2⁠</td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <th>6 </th> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠61/2⁠</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr></tbody></table></dd></dl> From the second, the numerators of the first column are the denominators of Euler's formula. The first column is −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ × <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A163982" class="extiw" title="oeis:A163982">A163982</a>. <div class="mw-heading mw-heading2"><h2 id="An_algorithmic_view:_the_Seidel_triangle">An algorithmic view: the Seidel triangle</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=27" title="Edit section: An algorithmic view: the Seidel triangle">edit</a>]</div> The sequence Sn has another unexpected yet important property: The denominators of Sn+1 divide the factorial n!. In other words: the numbers Tn = Sn + 1 n!, sometimes called <a href="/wiki/Alternating_permutations" class="mw-redirect" title="Alternating permutations">Euler zigzag numbers</a>, are integers. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}=1,\,1,\,1,\,2,\,5,\,16,\,61,\,272,\,1385,\,7936,\,50521,\,353792,\ldots \quad n=0,1,2,3,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>5</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>16</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>61</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>272</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>1385</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>7936</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>50521</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>353792</mn> <mo>,</mo> <mo>…</mo> <mspace width="1em" /> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{n}=1,\,1,\,1,\,2,\,5,\,16,\,61,\,272,\,1385,\,7936,\,50521,\,353792,\ldots \quad n=0,1,2,3,\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d751c387569fe39148ac96f17b11adae5046b389" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:79.811ex; height:2.509ex;" alt="{\displaystyle T_{n}=1,\,1,\,1,\,2,\,5,\,16,\,61,\,272,\,1385,\,7936,\,50521,\,353792,\ldots \quad n=0,1,2,3,\ldots }"> (<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A000111" class="extiw" title="oeis:A000111">A000111</a>). See (<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A253671" class="extiw" title="oeis:A253671">A253671</a>).</dd></dl> Their <a href="/wiki/Exponential_generating_function" class="mw-redirect" title="Exponential generating function">exponential generating function</a> is the sum of the <a href="/wiki/Trigonometric_functions#Reciprocal_functions" title="Trigonometric functions">secant</a> and <a href="/wiki/Tangent_(trigonometry)" class="mw-redirect" title="Tangent (trigonometry)">tangent</a> functions. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }T_{n}{\frac {x^{n}}{n!}}=\tan \left({\frac {\pi }{4}}+{\frac {x}{2}}\right)=\sec x+\tan x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }T_{n}{\frac {x^{n}}{n!}}=\tan \left({\frac {\pi }{4}}+{\frac {x}{2}}\right)=\sec x+\tan x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e660d25c111b6bdc210e00fbf468e52253d15e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.823ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }T_{n}{\frac {x^{n}}{n!}}=\tan \left({\frac {\pi }{4}}+{\frac {x}{2}}\right)=\sec x+\tan x}">.</dd></dl> Thus the above representations of the Bernoulli and Euler numbers can be rewritten in terms of this sequence as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}B_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }[n{\text{ even}}]{\frac {n}{2^{n}-4^{n}}}\,T_{n-1}\ &n&\geq 2\\E_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }[n{\text{ even}}]T_{n}&n&\geq 0\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mrow> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> even</mtext> </mrow> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mtext> </mtext> </mtd> <mtd> <mi>n</mi> </mtd> <mtd> <mi></mi> <mo>≥</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>⌋</mo> </mrow> </mrow> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> even</mtext> </mrow> <mo stretchy="false">]</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi>n</mi> </mtd> <mtd> <mi></mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}B_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }[n{\text{ even}}]{\frac {n}{2^{n}-4^{n}}}\,T_{n-1}\ &n&\geq 2\\E_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }[n{\text{ even}}]T_{n}&n&\geq 0\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc3fa19f015a506252e543380c68bc0c3342495" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:47.455ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}B_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }[n{\text{ even}}]{\frac {n}{2^{n}-4^{n}}}\,T_{n-1}\ &n&\geq 2\\E_{n}&=(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }[n{\text{ even}}]T_{n}&n&\geq 0\end{aligned}}}"></dd></dl> These identities make it easy to compute the Bernoulli and Euler numbers: the Euler numbers E2n are given immediately by T2n and the Bernoulli numbers B2n are fractions obtained from T2n - 1 by some easy shifting, avoiding rational arithmetic. What remains is to find a convenient way to compute the numbers Tn. However, already in 1877 <a href="/wiki/Philipp_Ludwig_von_Seidel" title="Philipp Ludwig von Seidel">Philipp Ludwig von Seidel</a> published an ingenious algorithm, which makes it simple to calculate Tn.<a href="#cite_note-Seidel1877-37">[34]</a> <div class="thumb tnone" style=""><div class="thumbinner" style="width:-moz-fit-content; width:fit-content;"><div class="thumbimage noresize" style="width:auto;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{crrrcc}{}&{}&{\color {red}1}&{}&{}&{}\\{}&{\rightarrow }&{\color {blue}1}&{\color {red}1}&{}\\{}&{\color {red}2}&{\color {blue}2}&{\color {blue}1}&{\leftarrow }\\{\rightarrow }&{\color {blue}2}&{\color {blue}4}&{\color {blue}5}&{\color {red}5}\\{\color {red}16}&{\color {blue}16}&{\color {blue}14}&{\color {blue}10}&{\color {blue}5}&{\leftarrow }\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center right right right center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>1</mn> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">→</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mn>1</mn> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>1</mn> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>2</mn> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mn>2</mn> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mn>1</mn> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">←</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">→</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mn>2</mn> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mn>4</mn> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mn>5</mn> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>5</mn> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mn>16</mn> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mn>16</mn> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mn>14</mn> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mn>10</mn> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="blue"> <mn>5</mn> </mstyle> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">←</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{crrrcc}{}&{}&{\color {red}1}&{}&{}&{}\\{}&{\rightarrow }&{\color {blue}1}&{\color {red}1}&{}\\{}&{\color {red}2}&{\color {blue}2}&{\color {blue}1}&{\leftarrow }\\{\rightarrow }&{\color {blue}2}&{\color {blue}4}&{\color {blue}5}&{\color {red}5}\\{\color {red}16}&{\color {blue}16}&{\color {blue}14}&{\color {blue}10}&{\color {blue}5}&{\leftarrow }\end{array}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f11c262ba67a0d0a93c132bac0ebe44c648c2a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:26.312ex; height:15.843ex;" alt="{\displaystyle {\begin{array}{crrrcc}{}&{}&{\color {red}1}&{}&{}&{}\\{}&{\rightarrow }&{\color {blue}1}&{\color {red}1}&{}\\{}&{\color {red}2}&{\color {blue}2}&{\color {blue}1}&{\leftarrow }\\{\rightarrow }&{\color {blue}2}&{\color {blue}4}&{\color {blue}5}&{\color {red}5}\\{\color {red}16}&{\color {blue}16}&{\color {blue}14}&{\color {blue}10}&{\color {blue}5}&{\leftarrow }\end{array}}}"></div><div class="thumbcaption">Seidel's algorithm for Tn</div></div></div> <ol><li>Start by putting 1 in row 0 and let k denote the number of the row currently being filled</li> <li>If k is odd, then put the number on the left end of the row k − 1 in the first position of the row k, and fill the row from the left to the right, with every entry being the sum of the number to the left and the number to the upper</li> <li>At the end of the row duplicate the last number.</li> <li>If k is even, proceed similar in the other direction.</li></ol> Seidel's algorithm is in fact much more general (see the exposition of Dominique Dumont <a href="#cite_note-Dumont1981-38">[35]</a>) and was rediscovered several times thereafter. Similar to Seidel's approach D. E. Knuth and T. J. Buckholtz gave a recurrence equation for the numbers T2n and recommended this method for computing B2n and E2n 'on electronic computers using only simple operations on integers'.<a href="#cite_note-KnuthBuckholtz1967-39">[36]</a> V. I. Arnold<a href="#cite_note-Arnold1991-40">[37]</a> rediscovered Seidel's algorithm and later Millar, Sloane and Young popularized Seidel's algorithm under the name <a href="/wiki/Boustrophedon_transform" title="Boustrophedon transform">boustrophedon transform</a>. Triangular form: <dl><dd><table style="text-align:right"> <tbody><tr> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td>1</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <td></td> <td></td> <td></td> <td></td> <td></td> <td>1</td> <td></td> <td>1</td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <td></td> <td></td> <td></td> <td></td> <td>2</td> <td></td> <td>2</td> <td></td> <td>1</td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <td></td> <td></td> <td></td> <td>2</td> <td></td> <td>4</td> <td></td> <td>5</td> <td></td> <td>5</td> <td></td> <td></td> <td> </td></tr> <tr> <td></td> <td></td> <td>16</td> <td></td> <td>16</td> <td></td> <td>14</td> <td></td> <td>10</td> <td></td> <td>5</td> <td></td> <td> </td></tr> <tr> <td></td> <td>16</td> <td></td> <td>32</td> <td></td> <td>46</td> <td></td> <td>56</td> <td></td> <td>61</td> <td></td> <td>61</td> <td> </td></tr> <tr> <td>272</td> <td></td> <td>272</td> <td></td> <td>256</td> <td></td> <td>224</td> <td></td> <td>178</td> <td></td> <td>122</td> <td></td> <td>61 </td></tr></tbody></table></dd></dl> Only <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A000657" class="extiw" title="oeis:A000657">A000657</a>, with one 1, and <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A214267" class="extiw" title="oeis:A214267">A214267</a>, with two 1s, are in the OEIS. Distribution with a supplementary 1 and one 0 in the following rows: <dl><dd><table style="text-align:right"> <tbody><tr> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td>1</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <td></td> <td></td> <td></td> <td></td> <td></td> <td>0</td> <td></td> <td>1</td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <td></td> <td></td> <td></td> <td></td> <td>−1</td> <td></td> <td>−1</td> <td></td> <td>0</td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <td></td> <td></td> <td></td> <td>0</td> <td></td> <td>−1</td> <td></td> <td>−2</td> <td></td> <td>−2</td> <td></td> <td></td> <td> </td></tr> <tr> <td></td> <td></td> <td>5</td> <td></td> <td>5</td> <td></td> <td>4</td> <td></td> <td>2</td> <td></td> <td>0</td> <td></td> <td> </td></tr> <tr> <td></td> <td>0</td> <td></td> <td>5</td> <td></td> <td>10</td> <td></td> <td>14</td> <td></td> <td>16</td> <td></td> <td>16</td> <td> </td></tr> <tr> <td>−61</td> <td></td> <td>−61</td> <td></td> <td>−56</td> <td></td> <td>−46</td> <td></td> <td>−32</td> <td></td> <td>−16</td> <td></td> <td>0 </td></tr></tbody></table></dd></dl> This is <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A239005" class="extiw" title="oeis:A239005">A239005</a>, a signed version of <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A008280" class="extiw" title="oeis:A008280">A008280</a>. The main andiagonal is <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A122045" class="extiw" title="oeis:A122045">A122045</a>. The main diagonal is <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A155585" class="extiw" title="oeis:A155585">A155585</a>. The central column is <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A099023" class="extiw" title="oeis:A099023">A099023</a>. Row sums: 1, 1, −2, −5, 16, 61.... See <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A163747" class="extiw" title="oeis:A163747">A163747</a>. See the array beginning with 1, 1, 0, −2, 0, 16, 0 below. The Akiyama–Tanigawa algorithm applied to <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A046978" class="extiw" title="oeis:A046978">A046978</a> (n + 1) / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A016116" class="extiw" title="oeis:A016116">A016116</a>(n) yields: <dl><dd><table style="text-align:right"> <tbody><tr> <td>1</td> <td>1</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠</td> <td>0</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/8⁠ </td></tr> <tr> <td>0</td> <td>1</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/2⁠</td> <td>1</td> <td>0</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/4⁠ </td></tr> <tr> <td>−1</td> <td>−1</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/2⁠</td> <td>4</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠15/4⁠ </td></tr> <tr> <td>0</td> <td>−5</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠15/2⁠</td> <td>1 </td></tr> <tr> <td>5</td> <td>5</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠51/2⁠ </td></tr> <tr> <td>0</td> <td>61 </td></tr> <tr> <td>−61 </td></tr></tbody></table></dd></dl> 1. The first column is <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A122045" class="extiw" title="oeis:A122045">A122045</a>. Its binomial transform leads to: <dl><dd><table style="text-align:right"> <tbody><tr> <td>1</td> <td>1</td> <td>0</td> <td>−2</td> <td>0</td> <td>16</td> <td>0 </td></tr> <tr> <td>0</td> <td>−1</td> <td>−2</td> <td>2</td> <td>16</td> <td>−16 </td></tr> <tr> <td>−1</td> <td>−1</td> <td>4</td> <td>14</td> <td>−32 </td></tr> <tr> <td>0</td> <td>5</td> <td>10</td> <td>−46 </td></tr> <tr> <td>5</td> <td>5</td> <td>−56 </td></tr> <tr> <td>0</td> <td>−61 </td></tr> <tr> <td>−61 </td></tr></tbody></table></dd></dl> The first row of this array is <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A155585" class="extiw" title="oeis:A155585">A155585</a>. The absolute values of the increasing antidiagonals are <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A008280" class="extiw" title="oeis:A008280">A008280</a>. The sum of the antidiagonals is −<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A163747" class="extiw" title="oeis:A163747">A163747</a> (n + 1). 2. The second column is 1 1 −1 −5 5 61 −61 −1385 1385.... Its binomial transform yields: <dl><dd><table style="text-align:right"> <tbody><tr> <td>1</td> <td>2</td> <td>2</td> <td>−4</td> <td>−16</td> <td>32</td> <td>272 </td></tr> <tr> <td>1</td> <td>0</td> <td>−6</td> <td>−12</td> <td>48</td> <td>240 </td></tr> <tr> <td>−1</td> <td>−6</td> <td>−6</td> <td>60</td> <td>192 </td></tr> <tr> <td>−5</td> <td>0</td> <td>66</td> <td>32 </td></tr> <tr> <td>5</td> <td>66</td> <td>66 </td></tr> <tr> <td>61</td> <td>0 </td></tr> <tr> <td>−61 </td></tr></tbody></table></dd></dl> The first row of this array is 1 2 2 −4 −16 32 272 544 −7936 15872 353792 −707584.... The absolute values of the second bisection are the double of the absolute values of the first bisection. Consider the Akiyama-Tanigawa algorithm applied to <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A046978" class="extiw" title="oeis:A046978">A046978</a> (n) / (<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A158780" class="extiw" title="oeis:A158780">A158780</a> (n + 1) = abs(<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A117575" class="extiw" title="oeis:A117575">A117575</a> (n)) + 1 = 1, 2, 2, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/2⁠, 1, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/4⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/4⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠7/8⁠, 1, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠17/16⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠17/16⁠, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠33/32⁠.... <dl><dd><table style="text-align:right"> <tbody><tr> <td>1</td> <td>2</td> <td>2</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/2⁠</td> <td>1</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/4⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/4⁠ </td></tr> <tr> <td>−1</td> <td>0</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/2⁠</td> <td>2</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/4⁠</td> <td>0 </td></tr> <tr> <td>−1</td> <td>−3</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/2⁠</td> <td>3</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠25/4⁠ </td></tr> <tr> <td>2</td> <td>−3</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠27/2⁠</td> <td>−13 </td></tr> <tr> <td>5</td> <td>21</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/2⁠ </td></tr> <tr> <td>−16</td> <td>45 </td></tr> <tr> <td>−61 </td></tr></tbody></table></dd></dl> The first column whose the absolute values are <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A000111" class="extiw" title="oeis:A000111">A000111</a> could be the numerator of a trigonometric function. <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A163747" class="extiw" title="oeis:A163747">A163747</a> is an autosequence of the first kind (the main diagonal is <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A000004" class="extiw" title="oeis:A000004">A000004</a>). The corresponding array is: <dl><dd><table style="text-align:right"> <tbody><tr> <td>0</td> <td>−1</td> <td>−1</td> <td>2</td> <td>5</td> <td>−16</td> <td>−61 </td></tr> <tr> <td>−1</td> <td>0</td> <td>3</td> <td>3</td> <td>−21</td> <td>−45 </td></tr> <tr> <td>1</td> <td>3</td> <td>0</td> <td>−24</td> <td>−24 </td></tr> <tr> <td>2</td> <td>−3</td> <td>−24</td> <td>0 </td></tr> <tr> <td>−5</td> <td>−21</td> <td>24 </td></tr> <tr> <td>−16</td> <td>45 </td></tr> <tr> <td>−61 </td></tr></tbody></table></dd></dl> The first two upper diagonals are −1 3 −24 402... = (−1)n + 1 × <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A002832" class="extiw" title="oeis:A002832">A002832</a>. The sum of the antidiagonals is 0 −2 0 10... = 2 × <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A122045" class="extiw" title="oeis:A122045">A122045</a>(n + 1). −<a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A163982" class="extiw" title="oeis:A163982">A163982</a> is an autosequence of the second kind, like for instance <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A164555" class="extiw" title="oeis:A164555">A164555</a> / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A027642" class="extiw" title="oeis:A027642">A027642</a>. Hence the array: <dl><dd><table style="text-align:right"> <tbody><tr> <td>2</td> <td>1</td> <td>−1</td> <td>−2</td> <td>5</td> <td>16</td> <td>−61 </td></tr> <tr> <td>−1</td> <td>−2</td> <td>−1</td> <td>7</td> <td>11</td> <td>−77 </td></tr> <tr> <td>−1</td> <td>1</td> <td>8</td> <td>4</td> <td>−88 </td></tr> <tr> <td>2</td> <td>7</td> <td>−4</td> <td>−92 </td></tr> <tr> <td>5</td> <td>−11</td> <td>−88 </td></tr> <tr> <td>−16</td> <td>−77 </td></tr> <tr> <td>−61 </td></tr></tbody></table></dd></dl> The main diagonal, here 2 −2 8 −92..., is the double of the first upper one, here <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A099023" class="extiw" title="oeis:A099023">A099023</a>. The sum of the antidiagonals is 2 0 −4 0... = 2 × <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A155585" class="extiw" title="oeis:A155585">A155585</a>(n + 1). <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A163747" class="extiw" title="oeis:A163747">A163747</a> − <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A163982" class="extiw" title="oeis:A163982">A163982</a> = 2 × <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A122045" class="extiw" title="oeis:A122045">A122045</a>. <div class="mw-heading mw-heading2"><h2 id="A_combinatorial_view:_alternating_permutations">A combinatorial view: alternating permutations</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=28" title="Edit section: A combinatorial view: alternating permutations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Alternating_permutations" class="mw-redirect" title="Alternating permutations">Alternating permutations</a></div> Around 1880, three years after the publication of Seidel's algorithm, <a href="/wiki/D%C3%A9sir%C3%A9_Andr%C3%A9" title="Désiré André">Désiré André</a> proved a now classic result of combinatorial analysis.<a href="#cite_note-André1879-41">[38]</a><a href="#cite_note-André1881-42">[39]</a> Looking at the first terms of the Taylor expansion of the <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> tan x and sec x André made a startling discovery. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\tan x&=x+{\frac {2x^{3}}{3!}}+{\frac {16x^{5}}{5!}}+{\frac {272x^{7}}{7!}}+{\frac {7936x^{9}}{9!}}+\cdots \\[6pt]\sec x&=1+{\frac {x^{2}}{2!}}+{\frac {5x^{4}}{4!}}+{\frac {61x^{6}}{6!}}+{\frac {1385x^{8}}{8!}}+{\frac {50521x^{10}}{10!}}+\cdots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>16</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>272</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mrow> <mn>7</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>7936</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> </mrow> <mrow> <mn>9</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mi>sec</mi> <mo>⁡</mo> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>61</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> <mrow> <mn>6</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1385</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>50521</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> </mrow> <mrow> <mn>10</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan x&=x+{\frac {2x^{3}}{3!}}+{\frac {16x^{5}}{5!}}+{\frac {272x^{7}}{7!}}+{\frac {7936x^{9}}{9!}}+\cdots \\[6pt]\sec x&=1+{\frac {x^{2}}{2!}}+{\frac {5x^{4}}{4!}}+{\frac {61x^{6}}{6!}}+{\frac {1385x^{8}}{8!}}+{\frac {50521x^{10}}{10!}}+\cdots \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac611c1e77f2bfe7df44e140ba074784ba57c52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:60.726ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\tan x&=x+{\frac {2x^{3}}{3!}}+{\frac {16x^{5}}{5!}}+{\frac {272x^{7}}{7!}}+{\frac {7936x^{9}}{9!}}+\cdots \\[6pt]\sec x&=1+{\frac {x^{2}}{2!}}+{\frac {5x^{4}}{4!}}+{\frac {61x^{6}}{6!}}+{\frac {1385x^{8}}{8!}}+{\frac {50521x^{10}}{10!}}+\cdots \end{aligned}}}"></dd></dl> The coefficients are the <a href="/wiki/Euler_number" class="mw-redirect" title="Euler number">Euler numbers</a> of odd and even index, respectively. In consequence the ordinary expansion of tan x + sec x has as coefficients the rational numbers Sn. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan x+\sec x=1+x+{\tfrac {1}{2}}x^{2}+{\tfrac {1}{3}}x^{3}+{\tfrac {5}{24}}x^{4}+{\tfrac {2}{15}}x^{5}+{\tfrac {61}{720}}x^{6}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>sec</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>24</mn> </mfrac> </mstyle> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>15</mn> </mfrac> </mstyle> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>61</mn> <mn>720</mn> </mfrac> </mstyle> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan x+\sec x=1+x+{\tfrac {1}{2}}x^{2}+{\tfrac {1}{3}}x^{3}+{\tfrac {5}{24}}x^{4}+{\tfrac {2}{15}}x^{5}+{\tfrac {61}{720}}x^{6}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fccbd5d9a41683426710cfbaedb379fc36592a46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:64.309ex; height:3.676ex;" alt="{\displaystyle \tan x+\sec x=1+x+{\tfrac {1}{2}}x^{2}+{\tfrac {1}{3}}x^{3}+{\tfrac {5}{24}}x^{4}+{\tfrac {2}{15}}x^{5}+{\tfrac {61}{720}}x^{6}+\cdots }"></dd></dl> André then succeeded by means of a recurrence argument to show that the <a href="/wiki/Alternating_permutation" title="Alternating permutation">alternating permutations</a> of odd size are enumerated by the Euler numbers of odd index (also called tangent numbers) and the alternating permutations of even size by the Euler numbers of even index (also called secant numbers). <div class="mw-heading mw-heading2"><h2 id="Related_sequences">Related sequences</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=29" title="Edit section: Related sequences">edit</a>]</div> The arithmetic mean of the first and the second Bernoulli numbers are the associate Bernoulli numbers: B0 = 1, B1 = 0, B2 = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠, B3 = 0, B4 = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/30⁠, <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A176327" class="extiw" title="oeis:A176327">A176327</a> / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A027642" class="extiw" title="oeis:A027642">A027642</a>. Via the second row of its inverse Akiyama–Tanigawa transform <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A177427" class="extiw" title="oeis:A177427">A177427</a>, they lead to Balmer series <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A061037" class="extiw" title="oeis:A061037">A061037</a> / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A061038" class="extiw" title="oeis:A061038">A061038</a>. The Akiyama–Tanigawa algorithm applied to <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A060819" class="extiw" title="oeis:A060819">A060819</a> (n + 4) / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A145979" class="extiw" title="oeis:A145979">A145979</a> (n) leads to the Bernoulli numbers <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A027641" class="extiw" title="oeis:A027641">A027641</a> / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A027642" class="extiw" title="oeis:A027642">A027642</a>, <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A164555" class="extiw" title="oeis:A164555">A164555</a> / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A027642" class="extiw" title="oeis:A027642">A027642</a>, or <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A176327" class="extiw" title="oeis:A176327">A176327</a> <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A176289" class="extiw" title="oeis:A176289">A176289</a> without B1, named intrinsic Bernoulli numbers Bi(n). <dl><dd><table style="text-align:center; padding-left; padding-right: 2em;"> <tbody><tr> <td>1</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/6⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/4⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠7/10⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/3⁠ </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/20⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/15⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/42⁠ </td></tr> <tr> <td>0</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/30⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/20⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/35⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/84⁠ </td></tr> <tr> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/30⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/30⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/140⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/105⁠</td> <td>0 </td></tr> <tr> <td>0</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/42⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/28⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠4/105⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/28⁠ </td></tr></tbody></table></dd></dl> Hence another link between the intrinsic Bernoulli numbers and the Balmer series via <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A145979" class="extiw" title="oeis:A145979">A145979</a> (n). <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A145979" class="extiw" title="oeis:A145979">A145979</a> (n − 2) = 0, 2, 1, 6,... is a permutation of the non-negative numbers. The terms of the first row are f(n) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/n + 2⁠. 2, f(n) is an autosequence of the second kind. 3/2, f(n) leads by its inverse binomial transform to 3/2 −1/2 1/3 −1/4 1/5 ... = 1/2 + log 2. Consider g(n) = 1/2 – 1 / (n+2) = 0, 1/6, 1/4, 3/10, 1/3. The Akiyama-Tanagiwa transforms gives: <dl><dd><table style="text-align:center; padding-left; padding-right:2em;"> <tbody><tr> <td>0</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/10⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/14⁠</td> <td>... </td></tr> <tr> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/6⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/20⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/15⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/42⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/28⁠</td> <td>... </td></tr> <tr> <td>0</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/30⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/20⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠2/35⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/84⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/84⁠</td> <td>... </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/30⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/30⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/140⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/105⁠</td> <td>0</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/140⁠</td> <td>... </td></tr></tbody></table></dd></dl> 0, g(n), is an autosequence of the second kind. Euler <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A198631" class="extiw" title="oeis:A198631">A198631</a> (n) / <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A006519" class="extiw" title="oeis:A006519">A006519</a> (n + 1) without the second term (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠) are the fractional intrinsic Euler numbers Ei(n) = 1, 0, −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠, 0, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, 0, −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠17/8⁠, 0, ... The corresponding Akiyama transform is: <dl><dd><table style="text-align:center; padding-left; padding-right: 2em;"> <tbody><tr> <td>1</td> <td>1</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠7/8⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/4⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠21/32⁠ </td></tr> <tr> <td>0</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/8⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/8⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/16⁠ </td></tr> <tr> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td>0</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠25/64⁠ </td></tr> <tr> <td>0</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/4⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠9/16⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/32⁠ </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠9/16⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠13/8⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠125/64⁠ </td></tr></tbody></table></dd></dl> The first line is Eu(n). Eu(n) preceded by a zero is an autosequence of the first kind. It is linked to the Oresme numbers. The numerators of the second line are <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A069834" class="extiw" title="oeis:A069834">A069834</a> preceded by 0. The difference table is: <dl><dd><table style="text-align:center; padding-left; padding-right: 2em;"> <tbody><tr> <td>0</td> <td>1</td> <td>1</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠7/8⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/4⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠21/32⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠19/32⁠ </td></tr> <tr> <td>1</td> <td>0</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/8⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/8⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/32⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/16⁠</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/128⁠ </td></tr> <tr> <td>−1</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/8⁠</td> <td>0</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/32⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/32⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠3/128⁠</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/64⁠ </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Arithmetical_properties_of_the_Bernoulli_numbers">Arithmetical properties of the Bernoulli numbers</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=30" title="Edit section: Arithmetical properties of the Bernoulli numbers">edit</a>]</div> The Bernoulli numbers can be expressed in terms of the Riemann zeta function as Bn = −nζ(1 − n) for integers n ≥ 0 provided for n = 0 the expression −nζ(1 − n) is understood as the limiting value and the convention B1 = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠ is used. This intimately relates them to the values of the zeta function at negative integers. As such, they could be expected to have and do have deep arithmetical properties. For example, the <a href="/wiki/Agoh%E2%80%93Giuga_conjecture" title="Agoh–Giuga conjecture">Agoh–Giuga conjecture</a> postulates that p is a prime number if and only if pBp − 1 is congruent to −1 modulo p. Divisibility properties of the Bernoulli numbers are related to the <a href="/wiki/Ideal_class_group" title="Ideal class group">ideal class groups</a> of <a href="/wiki/Cyclotomic_field" title="Cyclotomic field">cyclotomic fields</a> by a theorem of Kummer and its strengthening in the <a href="/wiki/Herbrand-Ribet_theorem" class="mw-redirect" title="Herbrand-Ribet theorem">Herbrand-Ribet theorem</a>, and to class numbers of real quadratic fields by <a href="/wiki/Ankeny%E2%80%93Artin%E2%80%93Chowla_congruence" title="Ankeny–Artin–Chowla congruence">Ankeny–Artin–Chowla</a>. <div class="mw-heading mw-heading3"><h3 id="The_Kummer_theorems">The Kummer theorems</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=31" title="Edit section: The Kummer theorems">edit</a>]</div> The Bernoulli numbers are related to <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> (FLT) by <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Kummer</a>'s theorem,<a href="#cite_note-Kummer1850-43">[40]</a> which says: <dl><dd>If the odd prime p does not divide any of the numerators of the Bernoulli numbers B2, B4, ..., Bp − 3 then xp + yp + zp = 0 has no solutions in nonzero integers.</dd></dl> Prime numbers with this property are called <a href="/wiki/Regular_prime" title="Regular prime">regular primes</a>. Another classical result of Kummer are the following <a href="/wiki/Modular_arithmetic#Congruence" title="Modular arithmetic">congruences</a>.<a href="#cite_note-Kummer1851-44">[41]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Kummer%27s_congruence" title="Kummer's congruence">Kummer's congruence</a></div> <dl><dd>Let p be an odd prime and b an even number such that p − 1 does not divide b. Then for any non-negative integer k <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {B_{k(p-1)+b}}{k(p-1)+b}}\equiv {\frac {B_{b}}{b}}{\pmod {p}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> </mrow> </msub> <mrow> <mi>k</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mi>b</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {B_{k(p-1)+b}}{k(p-1)+b}}\equiv {\frac {B_{b}}{b}}{\pmod {p}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/597f2e73a3ba50f263b09ec86d829058899e77f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.003ex; height:6.509ex;" alt="{\displaystyle {\frac {B_{k(p-1)+b}}{k(p-1)+b}}\equiv {\frac {B_{b}}{b}}{\pmod {p}}.}"></dd></dl></dd></dl> A generalization of these congruences goes by the name of p-adic continuity. <div class="mw-heading mw-heading3"><h3 id="p-adic_continuity">p-adic continuity</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=32" title="Edit section: p-adic continuity">edit</a>]</div> If b, m and n are positive integers such that m and n are not divisible by p − 1 and m ≡ n (mod pb − 1 (p − 1)), then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-p^{m-1}){\frac {B_{m}}{m}}\equiv (1-p^{n-1}){\frac {B_{n}}{n}}{\pmod {p^{b}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>m</mi> </mfrac> </mrow> <mo>≡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>n</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-p^{m-1}){\frac {B_{m}}{m}}\equiv (1-p^{n-1}){\frac {B_{n}}{n}}{\pmod {p^{b}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982f69f12f67747600e19f7ed63627d883ea064f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.688ex; height:5.176ex;" alt="{\displaystyle (1-p^{m-1}){\frac {B_{m}}{m}}\equiv (1-p^{n-1}){\frac {B_{n}}{n}}{\pmod {p^{b}}}.}"></dd></dl> Since Bn = −nζ(1 − n), this can also be written <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(1-p^{-u}\right)\zeta (u)\equiv \left(1-p^{-v}\right)\zeta (v){\pmod {p^{b}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>u</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>v</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1-p^{-u}\right)\zeta (u)\equiv \left(1-p^{-v}\right)\zeta (v){\pmod {p^{b}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/989c7f68b043fe44daa7fcf38272b94910665ac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.94ex; height:3.343ex;" alt="{\displaystyle \left(1-p^{-u}\right)\zeta (u)\equiv \left(1-p^{-v}\right)\zeta (v){\pmod {p^{b}}},}"></dd></dl> where u = 1 − m and v = 1 − n, so that u and v are nonpositive and not congruent to 1 modulo p − 1. This tells us that the Riemann zeta function, with 1 − p−s taken out of the Euler product formula, is continuous in the <a href="/wiki/P-adic_number" title="P-adic number">p-adic numbers</a> on odd negative integers congruent modulo p − 1 to a particular a ≢ 1 mod (p − 1), and so can be extended to a continuous function ζp(s) for all p-adic integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7967f468b6a942f77dd96ada0815be530fcda626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.256ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p},}"> the <a href="/wiki/P-adic_zeta_function" class="mw-redirect" title="P-adic zeta function">p-adic zeta function</a>. <div class="mw-heading mw-heading3"><h3 id="Ramanujan's_congruences">Ramanujan's congruences</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=33" title="Edit section: Ramanujan's congruences">edit</a>]</div> The following relations, due to <a href="/wiki/Ramanujan" class="mw-redirect" title="Ramanujan">Ramanujan</a>, provide a method for calculating Bernoulli numbers that is more efficient than the one given by their original recursive definition: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {m+3}{m}}B_{m}={\begin{cases}{\frac {m+3}{3}}-\sum \limits _{j=1}^{\frac {m}{6}}{\binom {m+3}{m-6j}}B_{m-6j},&{\text{if }}m\equiv 0{\pmod {6}};\\{\frac {m+3}{3}}-\sum \limits _{j=1}^{\frac {m-2}{6}}{\binom {m+3}{m-6j}}B_{m-6j},&{\text{if }}m\equiv 2{\pmod {6}};\\-{\frac {m+3}{6}}-\sum \limits _{j=1}^{\frac {m-4}{6}}{\binom {m+3}{m-6j}}B_{m-6j},&{\text{if }}m\equiv 4{\pmod {6}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>3</mn> </mrow> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo>+</mo> <mn>3</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>6</mn> </mfrac> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>3</mn> </mrow> <mrow> <mi>m</mi> <mo>−</mo> <mn>6</mn> <mi>j</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>6</mn> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>m</mi> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo>+</mo> <mn>3</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo>−</mo> <mn>2</mn> </mrow> <mn>6</mn> </mfrac> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>3</mn> </mrow> <mrow> <mi>m</mi> <mo>−</mo> <mn>6</mn> <mi>j</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>6</mn> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>m</mi> <mo>≡</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo>+</mo> <mn>3</mn> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>−</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo>−</mo> <mn>4</mn> </mrow> <mn>6</mn> </mfrac> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>3</mn> </mrow> <mrow> <mi>m</mi> <mo>−</mo> <mn>6</mn> <mi>j</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>6</mn> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>m</mi> <mo>≡</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\binom {m+3}{m}}B_{m}={\begin{cases}{\frac {m+3}{3}}-\sum \limits _{j=1}^{\frac {m}{6}}{\binom {m+3}{m-6j}}B_{m-6j},&{\text{if }}m\equiv 0{\pmod {6}};\\{\frac {m+3}{3}}-\sum \limits _{j=1}^{\frac {m-2}{6}}{\binom {m+3}{m-6j}}B_{m-6j},&{\text{if }}m\equiv 2{\pmod {6}};\\-{\frac {m+3}{6}}-\sum \limits _{j=1}^{\frac {m-4}{6}}{\binom {m+3}{m-6j}}B_{m-6j},&{\text{if }}m\equiv 4{\pmod {6}}.\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f593dba092e4d79ff4c27ec6f256b4ad0ef2cc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.234ex; margin-bottom: -0.27ex; width:65.928ex; height:24.176ex;" alt="{\displaystyle {\binom {m+3}{m}}B_{m}={\begin{cases}{\frac {m+3}{3}}-\sum \limits _{j=1}^{\frac {m}{6}}{\binom {m+3}{m-6j}}B_{m-6j},&{\text{if }}m\equiv 0{\pmod {6}};\\{\frac {m+3}{3}}-\sum \limits _{j=1}^{\frac {m-2}{6}}{\binom {m+3}{m-6j}}B_{m-6j},&{\text{if }}m\equiv 2{\pmod {6}};\\-{\frac {m+3}{6}}-\sum \limits _{j=1}^{\frac {m-4}{6}}{\binom {m+3}{m-6j}}B_{m-6j},&{\text{if }}m\equiv 4{\pmod {6}}.\end{cases}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Von_Staudt–Clausen_theorem">Von Staudt–Clausen theorem</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=34" title="Edit section: Von Staudt–Clausen theorem">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Von_Staudt%E2%80%93Clausen_theorem" title="Von Staudt–Clausen theorem">Von Staudt–Clausen theorem</a></div> The von Staudt–Clausen theorem was given by <a href="/wiki/Karl_Georg_Christian_von_Staudt" title="Karl Georg Christian von Staudt">Karl Georg Christian von Staudt</a><a href="#cite_note-vonStaudt1840-45">[42]</a> and <a href="/wiki/Thomas_Clausen_(mathematician)" title="Thomas Clausen (mathematician)">Thomas Clausen</a><a href="#cite_note-Clausen1840-46">[43]</a> independently in 1840. The theorem states that for every n > 0, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{2n}+\sum _{(p-1)\,\mid \,2n}{\frac {1}{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>∣</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2n}+\sum _{(p-1)\,\mid \,2n}{\frac {1}{p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ccff12654cf0f82891558b62945018bcb42d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:16.284ex; height:6.843ex;" alt="{\displaystyle B_{2n}+\sum _{(p-1)\,\mid \,2n}{\frac {1}{p}}}"></dd></dl> is an integer. The sum extends over all <a href="/wiki/Prime_number" title="Prime number">primes</a> p for which p − 1 divides 2n. A consequence of this is that the denominator of B2n is given by the product of all primes p for which p − 1 divides 2n. In particular, these denominators are <a href="/wiki/Square-free" class="mw-redirect" title="Square-free">square-free</a> and divisible by 6. <div class="mw-heading mw-heading3"><h3 id="Why_do_the_odd_Bernoulli_numbers_vanish?">Why do the odd Bernoulli numbers vanish?</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=35" title="Edit section: Why do the odd Bernoulli numbers vanish?">edit</a>]</div> The sum <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{k}(n)=\sum _{i=0}^{n}i^{k}-{\frac {n^{k}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{k}(n)=\sum _{i=0}^{n}i^{k}-{\frac {n^{k}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b053fb2fc5f9ea916cc1fb73785d62fde4581bf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.705ex; height:6.843ex;" alt="{\displaystyle \varphi _{k}(n)=\sum _{i=0}^{n}i^{k}-{\frac {n^{k}}{2}}}"></dd></dl> can be evaluated for negative values of the index n. Doing so will show that it is an <a href="/wiki/Odd_function" class="mw-redirect" title="Odd function">odd function</a> for even values of k, which implies that the sum has only terms of odd index. This and the formula for the Bernoulli sum imply that B2k + 1 − m is 0 for m even and 2k + 1 − m > 1; and that the term for B1 is cancelled by the subtraction. The von Staudt–Clausen theorem combined with Worpitzky's representation also gives a combinatorial answer to this question (valid for n > 1). From the von Staudt–Clausen theorem it is known that for odd n > 1 the number 2Bn is an integer. This seems trivial if one knows beforehand that the integer in question is zero. However, by applying Worpitzky's representation one gets <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2B_{n}=\sum _{m=0}^{n}(-1)^{m}{\frac {2}{m+1}}m!\left\{{n+1 \atop m+1}\right\}=0\quad (n>1{\text{ is odd}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mi>m</mi> <mo>!</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> is odd</mtext> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2B_{n}=\sum _{m=0}^{n}(-1)^{m}{\frac {2}{m+1}}m!\left\{{n+1 \atop m+1}\right\}=0\quad (n>1{\text{ is odd}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ac9fe56907a0593bf2169527aaeb8582c4cfbac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.803ex; height:6.843ex;" alt="{\displaystyle 2B_{n}=\sum _{m=0}^{n}(-1)^{m}{\frac {2}{m+1}}m!\left\{{n+1 \atop m+1}\right\}=0\quad (n>1{\text{ is odd}})}"></dd></dl> as a sum of integers, which is not trivial. Here a combinatorial fact comes to surface which explains the vanishing of the Bernoulli numbers at odd index. Let Sn,m be the number of surjective maps from {1, 2, ..., n} to {1, 2, ..., m}, then Sn,m = m!<big><big>{</big></big>n m<big><big>}</big></big>. The last equation can only hold if <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{{\text{odd }}m=1}^{n-1}{\frac {2}{m^{2}}}S_{n,m}=\sum _{{\text{even }}m=2}^{n}{\frac {2}{m^{2}}}S_{n,m}\quad (n>2{\text{ is even}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>odd </mtext> </mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>even </mtext> </mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>></mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> is even</mtext> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{{\text{odd }}m=1}^{n-1}{\frac {2}{m^{2}}}S_{n,m}=\sum _{{\text{even }}m=2}^{n}{\frac {2}{m^{2}}}S_{n,m}\quad (n>2{\text{ is even}}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee9a8213639d83d1102c398d273102155fe7e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:52.705ex; height:7.343ex;" alt="{\displaystyle \sum _{{\text{odd }}m=1}^{n-1}{\frac {2}{m^{2}}}S_{n,m}=\sum _{{\text{even }}m=2}^{n}{\frac {2}{m^{2}}}S_{n,m}\quad (n>2{\text{ is even}}).}"></dd></dl> This equation can be proved by induction. The first two examples of this equation are <dl><dd>n = 4: 2 + 8 = 7 + 3,</dd> <dd>n = 6: 2 + 120 + 144 = 31 + 195 + 40.</dd></dl> Thus the Bernoulli numbers vanish at odd index because some non-obvious combinatorial identities are embodied in the Bernoulli numbers. <div class="mw-heading mw-heading3"><h3 id="A_restatement_of_the_Riemann_hypothesis">A restatement of the Riemann hypothesis</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=36" title="Edit section: A restatement of the Riemann hypothesis">edit</a>]</div> The connection between the Bernoulli numbers and the Riemann zeta function is strong enough to provide an alternate formulation of the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> (RH) which uses only the Bernoulli numbers. In fact <a href="/wiki/Marcel_Riesz" title="Marcel Riesz">Marcel Riesz</a> proved that the RH is equivalent to the following assertion:<a href="#cite_note-Riesz1916-47">[44]</a> <dl><dd>For every ε > <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/4⁠ there exists a constant Cε > 0 (depending on ε) such that |R(x)| < Cεxε as x → ∞.</dd></dl> Here R(x) is the <a href="/wiki/Riesz_function" title="Riesz function">Riesz function</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x)=2\sum _{k=1}^{\infty }{\frac {k^{\overline {k}}x^{k}}{(2\pi )^{2k}\left({\frac {B_{2k}}{2k}}\right)}}=2\sum _{k=1}^{\infty }{\frac {k^{\overline {k}}x^{k}}{(2\pi )^{2k}\beta _{2k}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo accent="false">¯</mo> </mover> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo accent="false">¯</mo> </mover> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(x)=2\sum _{k=1}^{\infty }{\frac {k^{\overline {k}}x^{k}}{(2\pi )^{2k}\left({\frac {B_{2k}}{2k}}\right)}}=2\sum _{k=1}^{\infty }{\frac {k^{\overline {k}}x^{k}}{(2\pi )^{2k}\beta _{2k}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a273b39acf1cd3bb9122272660668e545684a995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:46.431ex; height:8.843ex;" alt="{\displaystyle R(x)=2\sum _{k=1}^{\infty }{\frac {k^{\overline {k}}x^{k}}{(2\pi )^{2k}\left({\frac {B_{2k}}{2k}}\right)}}=2\sum _{k=1}^{\infty }{\frac {k^{\overline {k}}x^{k}}{(2\pi )^{2k}\beta _{2k}}}.}"></dd></dl> nk denotes the <a href="/wiki/Pochhammer_symbol#Alternate_notations" class="mw-redirect" title="Pochhammer symbol">rising factorial power</a> in the notation of <a href="/wiki/D._E._Knuth" class="mw-redirect" title="D. E. Knuth">D. E. Knuth</a>. The numbers βn = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠Bn/n⁠ occur frequently in the study of the zeta function and are significant because βn is a p-integer for primes p where p − 1 does not divide n. The βn are called divided Bernoulli numbers. <div class="mw-heading mw-heading2"><h2 id="Generalized_Bernoulli_numbers">Generalized Bernoulli numbers</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=37" title="Edit section: Generalized Bernoulli numbers">edit</a>]</div> The generalized Bernoulli numbers are certain <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a>, defined similarly to the Bernoulli numbers, that are related to <a href="/wiki/Special_values_of_L-functions" title="Special values of L-functions">special values</a> of <a href="/wiki/Dirichlet_L-function" title="Dirichlet L-function">Dirichlet L-functions</a> in the same way that Bernoulli numbers are related to special values of the Riemann zeta function. Let χ be a <a href="/wiki/Dirichlet_character" title="Dirichlet character">Dirichlet character</a> modulo f. The generalized Bernoulli numbers attached to χ are defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{a=1}^{f}\chi (a){\frac {te^{at}}{e^{ft}-1}}=\sum _{k=0}^{\infty }B_{k,\chi }{\frac {t^{k}}{k!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </munderover> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>t</mi> </mrow> </msup> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>χ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{a=1}^{f}\chi (a){\frac {te^{at}}{e^{ft}-1}}=\sum _{k=0}^{\infty }B_{k,\chi }{\frac {t^{k}}{k!}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de6d7cff17603e0e2d60765374a95a9abf5e4b6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.48ex; height:7.676ex;" alt="{\displaystyle \sum _{a=1}^{f}\chi (a){\frac {te^{at}}{e^{ft}-1}}=\sum _{k=0}^{\infty }B_{k,\chi }{\frac {t^{k}}{k!}}.}"></dd></dl> Apart from the exceptional B1,1 = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, we have, for any Dirichlet character χ, that Bk,χ = 0 if χ(−1) ≠ (−1)k. Generalizing the relation between Bernoulli numbers and values of the Riemann zeta function at non-positive integers, one has the for all integers k ≥ 1: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(1-k,\chi )=-{\frac {B_{k,\chi }}{k}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> <mo>,</mo> <mi>χ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>χ</mi> </mrow> </msub> <mi>k</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(1-k,\chi )=-{\frac {B_{k,\chi }}{k}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38568c794c03d79e6ebebbe25259860598c2bb64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.824ex; height:5.676ex;" alt="{\displaystyle L(1-k,\chi )=-{\frac {B_{k,\chi }}{k}},}"></dd></dl> where L(s,χ) is the Dirichlet L-function of χ.<a href="#cite_note-Neukirch1999_VII2-48">[45]</a> <div class="mw-heading mw-heading3"><h3 id="Eisenstein–Kronecker_number">Eisenstein–Kronecker number</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=38" title="Edit section: Eisenstein–Kronecker number">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Eisenstein%E2%80%93Kronecker_number" title="Eisenstein–Kronecker number">Eisenstein–Kronecker number</a></div> <a href="/wiki/Eisenstein%E2%80%93Kronecker_number" title="Eisenstein–Kronecker number">Eisenstein–Kronecker numbers</a> are an analogue of the generalized Bernoulli numbers for <a href="/wiki/Imaginary_quadratic_field" class="mw-redirect" title="Imaginary quadratic field">imaginary quadratic fields</a>.<a href="#cite_note-Charollois-Sczech-49">[46]</a><a href="#cite_note-BK-50">[47]</a> They are related to critical L-values of <a href="/wiki/Hecke_character" title="Hecke character">Hecke characters</a>.<a href="#cite_note-BK-50">[47]</a> <div class="mw-heading mw-heading2"><h2 id="Appendix">Appendix</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=39" title="Edit section: Appendix">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Assorted_identities">Assorted identities</h3>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=40" title="Edit section: Assorted identities">edit</a>]</div> <div><ul><li><a href="/wiki/Umbral_calculus" title="Umbral calculus">Umbral calculus</a> gives a compact form of Bernoulli's formula by using an abstract symbol B: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{m}(n)={\frac {1}{m+1}}((\mathbf {B} +n)^{m+1}-B_{m+1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>+</mo> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{m}(n)={\frac {1}{m+1}}((\mathbf {B} +n)^{m+1}-B_{m+1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd70631cb4fa5a491a5b79547fba277cb37dda3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:38.192ex; height:5.343ex;" alt="{\displaystyle S_{m}(n)={\frac {1}{m+1}}((\mathbf {B} +n)^{m+1}-B_{m+1})}"></dd></dl> where the symbol Bk that appears during binomial expansion of the parenthesized term is to be replaced by the Bernoulli number Bk (and B1 = +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠). More suggestively and mnemonically, this may be written as a definite integral: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{m}(n)=\int _{0}^{n}(\mathbf {B} +x)^{m}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{m}(n)=\int _{0}^{n}(\mathbf {B} +x)^{m}\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fabc12e6f0825fbc08fafaca57d1891666458c58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.573ex; height:5.843ex;" alt="{\displaystyle S_{m}(n)=\int _{0}^{n}(\mathbf {B} +x)^{m}\,dx}"></dd></dl> Many other Bernoulli identities can be written compactly with this symbol, e.g. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-2\mathbf {B} )^{m}=(2-2^{m})B_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-2\mathbf {B} )^{m}=(2-2^{m})B_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb3d893e547c1768aa7c35be70a3742b6d7eb1e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.738ex; height:2.843ex;" alt="{\displaystyle (1-2\mathbf {B} )^{m}=(2-2^{m})B_{m}}"></dd></dl></li><li>Let n be non-negative and even <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (n)={\frac {(-1)^{{\frac {n}{2}}-1}B_{n}(2\pi )^{n}}{2(n!)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (n)={\frac {(-1)^{{\frac {n}{2}}-1}B_{n}(2\pi )^{n}}{2(n!)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b788f3e07762f2d789845f47415f7d75e3469447" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.488ex; height:7.343ex;" alt="{\displaystyle \zeta (n)={\frac {(-1)^{{\frac {n}{2}}-1}B_{n}(2\pi )^{n}}{2(n!)}}}"></dd></dl></li><li>The nth <a href="/wiki/Cumulant" title="Cumulant">cumulant</a> of the <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform</a> <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> on the interval [−1, 0] is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠Bn/n⁠.</li><li>Let n? = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/n!⁠ and n ≥ 1. Then Bn is the following (n + 1) × (n + 1) determinant:<a href="#cite_note-Malenfant2011-51">[48]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}B_{n}&=n!{\begin{vmatrix}1&0&\cdots &0&1\\2?&1&\cdots &0&0\\\vdots &\vdots &&\vdots &\vdots \\n?&(n-1)?&\cdots &1&0\\(n+1)?&n?&\cdots &2?&0\end{vmatrix}}\\[8pt]&=n!{\begin{vmatrix}1&0&\cdots &0&1\\{\frac {1}{2!}}&1&\cdots &0&0\\\vdots &\vdots &&\vdots &\vdots \\{\frac {1}{n!}}&{\frac {1}{(n-1)!}}&\cdots &1&0\\{\frac {1}{(n+1)!}}&{\frac {1}{n!}}&\cdots &{\frac {1}{2!}}&0\end{vmatrix}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mo>?</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>n</mi> <mo>?</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>?</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>?</mo> </mtd> <mtd> <mi>n</mi> <mo>?</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>2</mn> <mo>?</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}B_{n}&=n!{\begin{vmatrix}1&0&\cdots &0&1\\2?&1&\cdots &0&0\\\vdots &\vdots &&\vdots &\vdots \\n?&(n-1)?&\cdots &1&0\\(n+1)?&n?&\cdots &2?&0\end{vmatrix}}\\[8pt]&=n!{\begin{vmatrix}1&0&\cdots &0&1\\{\frac {1}{2!}}&1&\cdots &0&0\\\vdots &\vdots &&\vdots &\vdots \\{\frac {1}{n!}}&{\frac {1}{(n-1)!}}&\cdots &1&0\\{\frac {1}{(n+1)!}}&{\frac {1}{n!}}&\cdots &{\frac {1}{2!}}&0\end{vmatrix}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeed24c819804a002d3b8e024d65b71203d61be0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.338ex; width:42.964ex; height:39.843ex;" alt="{\displaystyle {\begin{aligned}B_{n}&=n!{\begin{vmatrix}1&0&\cdots &0&1\\2?&1&\cdots &0&0\\\vdots &\vdots &&\vdots &\vdots \\n?&(n-1)?&\cdots &1&0\\(n+1)?&n?&\cdots &2?&0\end{vmatrix}}\\[8pt]&=n!{\begin{vmatrix}1&0&\cdots &0&1\\{\frac {1}{2!}}&1&\cdots &0&0\\\vdots &\vdots &&\vdots &\vdots \\{\frac {1}{n!}}&{\frac {1}{(n-1)!}}&\cdots &1&0\\{\frac {1}{(n+1)!}}&{\frac {1}{n!}}&\cdots &{\frac {1}{2!}}&0\end{vmatrix}}\end{aligned}}}"></dd></dl> Thus the determinant is σn(1), the <a href="/wiki/Stirling_polynomial" class="mw-redirect" title="Stirling polynomial">Stirling polynomial</a> at x = 1.</li><li>For even-numbered Bernoulli numbers, B2p is given by the (p + 1) × (p + 1) determinant::<a href="#cite_note-Malenfant2011-51">[48]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{2p}=-{\frac {(2p)!}{2^{2p}-2}}{\begin{vmatrix}1&0&0&\cdots &0&1\\{\frac {1}{3!}}&1&0&\cdots &0&0\\{\frac {1}{5!}}&{\frac {1}{3!}}&1&\cdots &0&0\\\vdots &\vdots &\vdots &&\vdots &\vdots \\{\frac {1}{(2p+1)!}}&{\frac {1}{(2p-1)!}}&{\frac {1}{(2p-3)!}}&\cdots &{\frac {1}{3!}}&0\end{vmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>p</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>p</mi> </mrow> </msup> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>5</mn> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>p</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2p}=-{\frac {(2p)!}{2^{2p}-2}}{\begin{vmatrix}1&0&0&\cdots &0&1\\{\frac {1}{3!}}&1&0&\cdots &0&0\\{\frac {1}{5!}}&{\frac {1}{3!}}&1&\cdots &0&0\\\vdots &\vdots &\vdots &&\vdots &\vdots \\{\frac {1}{(2p+1)!}}&{\frac {1}{(2p-1)!}}&{\frac {1}{(2p-3)!}}&\cdots &{\frac {1}{3!}}&0\end{vmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c974fe9a53a8aa6d22f425690c004435961a16a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:55.061ex; height:20.176ex;" alt="{\displaystyle B_{2p}=-{\frac {(2p)!}{2^{2p}-2}}{\begin{vmatrix}1&0&0&\cdots &0&1\\{\frac {1}{3!}}&1&0&\cdots &0&0\\{\frac {1}{5!}}&{\frac {1}{3!}}&1&\cdots &0&0\\\vdots &\vdots &\vdots &&\vdots &\vdots \\{\frac {1}{(2p+1)!}}&{\frac {1}{(2p-1)!}}&{\frac {1}{(2p-3)!}}&\cdots &{\frac {1}{3!}}&0\end{vmatrix}}}"></dd></dl></li><li>Let n ≥ 1. Then (<a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>)<a href="#cite_note-52">[49]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}{\binom {n}{k}}B_{k}B_{n-k}+B_{n-1}=-B_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}{\binom {n}{k}}B_{k}B_{n-k}+B_{n-1}=-B_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d888fa118fce215bb8294f6ed9069c3ba1de1264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.959ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}{\binom {n}{k}}B_{k}B_{n-k}+B_{n-1}=-B_{n}}"></dd></dl></li><li>Let n ≥ 1. Then<a href="#cite_note-vonEttingshausen1827-53">[50]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{n}{\binom {n+1}{k}}(n+k+1)B_{n+k}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{n}{\binom {n+1}{k}}(n+k+1)B_{n+k}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68dc2790b139a92efb1c90d0332a485aec85ffce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.198ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{n}{\binom {n+1}{k}}(n+k+1)B_{n+k}=0}"></dd></dl></li><li>Let n ≥ 0. Then (<a href="/wiki/Leopold_Kronecker" title="Leopold Kronecker">Leopold Kronecker</a> 1883) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}=-\sum _{k=1}^{n+1}{\frac {(-1)^{k}}{k}}{\binom {n+1}{k}}\sum _{j=1}^{k}j^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mi>k</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}=-\sum _{k=1}^{n+1}{\frac {(-1)^{k}}{k}}{\binom {n+1}{k}}\sum _{j=1}^{k}j^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f29725a93571492586849b04a06e43f20880637" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.847ex; height:7.676ex;" alt="{\displaystyle B_{n}=-\sum _{k=1}^{n+1}{\frac {(-1)^{k}}{k}}{\binom {n+1}{k}}\sum _{j=1}^{k}j^{n}}"></dd></dl></li><li>Let n ≥ 1 and m ≥ 1. Then<a href="#cite_note-Carlitz1968-54">[51]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{m}\sum _{r=0}^{m}{\binom {m}{r}}B_{n+r}=(-1)^{n}\sum _{s=0}^{n}{\binom {n}{s}}B_{m+s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>m</mi> <mi>r</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>s</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{m}\sum _{r=0}^{m}{\binom {m}{r}}B_{n+r}=(-1)^{n}\sum _{s=0}^{n}{\binom {n}{s}}B_{m+s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69776eff31afcb467449d0e8ca1d66785286ae39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.579ex; height:6.843ex;" alt="{\displaystyle (-1)^{m}\sum _{r=0}^{m}{\binom {m}{r}}B_{n+r}=(-1)^{n}\sum _{s=0}^{n}{\binom {n}{s}}B_{m+s}}"></dd></dl></li><li>Let n ≥ 4 and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}=\sum _{k=1}^{n}k^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}=\sum _{k=1}^{n}k^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/669142a34d4d94cf01539afc37e88e6e06d3bdd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.534ex; height:6.843ex;" alt="{\displaystyle H_{n}=\sum _{k=1}^{n}k^{-1}}"></dd></dl> the <a href="/wiki/Harmonic_number" title="Harmonic number">harmonic number</a>. Then (H. Miki 1978) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n}{2}}\sum _{k=2}^{n-2}{\frac {B_{n-k}}{n-k}}{\frac {B_{k}}{k}}-\sum _{k=2}^{n-2}{\binom {n}{k}}{\frac {B_{n-k}}{n-k}}B_{k}=H_{n}B_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>k</mi> </mfrac> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n}{2}}\sum _{k=2}^{n-2}{\frac {B_{n-k}}{n-k}}{\frac {B_{k}}{k}}-\sum _{k=2}^{n-2}{\binom {n}{k}}{\frac {B_{n-k}}{n-k}}B_{k}=H_{n}B_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5746fc7c367307e7cfcd02a7850108cd9177bd90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.096ex; height:7.343ex;" alt="{\displaystyle {\frac {n}{2}}\sum _{k=2}^{n-2}{\frac {B_{n-k}}{n-k}}{\frac {B_{k}}{k}}-\sum _{k=2}^{n-2}{\binom {n}{k}}{\frac {B_{n-k}}{n-k}}B_{k}=H_{n}B_{n}}"></dd></dl></li><li>Let n ≥ 4. <a href="/wiki/Yuri_Matiyasevich" title="Yuri Matiyasevich">Yuri Matiyasevich</a> found (1997) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+2)\sum _{k=2}^{n-2}B_{k}B_{n-k}-2\sum _{l=2}^{n-2}{\binom {n+2}{l}}B_{l}B_{n-l}=n(n+1)B_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </munderover> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> <mi>l</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n+2)\sum _{k=2}^{n-2}B_{k}B_{n-k}-2\sum _{l=2}^{n-2}{\binom {n+2}{l}}B_{l}B_{n-l}=n(n+1)B_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cec8b6d9980f9babddf0298506e23da793c593e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.177ex; height:7.343ex;" alt="{\displaystyle (n+2)\sum _{k=2}^{n-2}B_{k}B_{n-k}-2\sum _{l=2}^{n-2}{\binom {n+2}{l}}B_{l}B_{n-l}=n(n+1)B_{n}}"></dd></dl></li><li>Faber–<a href="/wiki/Rahul_Pandharipande" title="Rahul Pandharipande">Pandharipande</a>–<a href="/wiki/Zagier" class="mw-redirect" title="Zagier">Zagier</a>–Gessel identity: for n ≥ 1, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n}{2}}\left(B_{n-1}(x)+\sum _{k=1}^{n-1}{\frac {B_{k}(x)}{k}}{\frac {B_{n-k}(x)}{n-k}}\right)-\sum _{k=0}^{n-1}{\binom {n}{k}}{\frac {B_{n-k}}{n-k}}B_{k}(x)=H_{n-1}B_{n}(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n}{2}}\left(B_{n-1}(x)+\sum _{k=1}^{n-1}{\frac {B_{k}(x)}{k}}{\frac {B_{n-k}(x)}{n-k}}\right)-\sum _{k=0}^{n-1}{\binom {n}{k}}{\frac {B_{n-k}}{n-k}}B_{k}(x)=H_{n-1}B_{n}(x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b2c73cbce1c0132ece9f7b239475608156eb89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:75.814ex; height:7.509ex;" alt="{\displaystyle {\frac {n}{2}}\left(B_{n-1}(x)+\sum _{k=1}^{n-1}{\frac {B_{k}(x)}{k}}{\frac {B_{n-k}(x)}{n-k}}\right)-\sum _{k=0}^{n-1}{\binom {n}{k}}{\frac {B_{n-k}}{n-k}}B_{k}(x)=H_{n-1}B_{n}(x).}"></dd></dl> Choosing x = 0 or x = 1 results in the Bernoulli number identity in one or another convention.</li><li>The next formula is true for n ≥ 0 if B1 = B1(1) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠, but only for n ≥ 1 if B1 = B1(0) = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/2⁠. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{n}{\binom {n}{k}}{\frac {B_{k}}{n-k+2}}={\frac {B_{n+1}}{n+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{n}{\binom {n}{k}}{\frac {B_{k}}{n-k+2}}={\frac {B_{n+1}}{n+1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6d5f5e93c0dcc8ca06fe900df52363c2f40460a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.175ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{n}{\binom {n}{k}}{\frac {B_{k}}{n-k+2}}={\frac {B_{n+1}}{n+1}}}"></dd></dl></li><li>Let n ≥ 0. Then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1+\sum _{k=0}^{n}{\binom {n}{k}}{\frac {2^{n-k+1}}{n-k+1}}B_{k}(1)=2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1+\sum _{k=0}^{n}{\binom {n}{k}}{\frac {2^{n-k+1}}{n-k+1}}B_{k}(1)=2^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6be874c489a5dae12dd54ede210484b1fb42bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.958ex; height:7.009ex;" alt="{\displaystyle -1+\sum _{k=0}^{n}{\binom {n}{k}}{\frac {2^{n-k+1}}{n-k+1}}B_{k}(1)=2^{n}}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1+\sum _{k=0}^{n}{\binom {n}{k}}{\frac {2^{n-k+1}}{n-k+1}}B_{k}(0)=\delta _{n,0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1+\sum _{k=0}^{n}{\binom {n}{k}}{\frac {2^{n-k+1}}{n-k+1}}B_{k}(0)=\delta _{n,0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bede6d7bad33c343b7f761126e43dd5b1a9846f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.107ex; height:7.009ex;" alt="{\displaystyle -1+\sum _{k=0}^{n}{\binom {n}{k}}{\frac {2^{n-k+1}}{n-k+1}}B_{k}(0)=\delta _{n,0}}"></dd></dl></li><li>A reciprocity relation of M. B. Gelfand:<a href="#cite_note-AgohDilcher2008-55">[52]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{m+1}\sum _{j=0}^{k}{\binom {k}{j}}{\frac {B_{m+1+j}}{m+1+j}}+(-1)^{k+1}\sum _{j=0}^{m}{\binom {m}{j}}{\frac {B_{k+1+j}}{k+1+j}}={\frac {k!m!}{(k+m+1)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>k</mi> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> </msub> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>m</mi> <mi>j</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> </msub> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>!</mo> <mi>m</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{m+1}\sum _{j=0}^{k}{\binom {k}{j}}{\frac {B_{m+1+j}}{m+1+j}}+(-1)^{k+1}\sum _{j=0}^{m}{\binom {m}{j}}{\frac {B_{k+1+j}}{k+1+j}}={\frac {k!m!}{(k+m+1)!}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7218f61980ace5d588fdf018d6675a17def4c479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:74.729ex; height:7.676ex;" alt="{\displaystyle (-1)^{m+1}\sum _{j=0}^{k}{\binom {k}{j}}{\frac {B_{m+1+j}}{m+1+j}}+(-1)^{k+1}\sum _{j=0}^{m}{\binom {m}{j}}{\frac {B_{k+1+j}}{k+1+j}}={\frac {k!m!}{(k+m+1)!}}}"></dd></dl></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=41" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Bernoulli_polynomial" class="mw-redirect" title="Bernoulli polynomial">Bernoulli polynomial</a></li> <li><a href="/wiki/Bernoulli_polynomials_of_the_second_kind" title="Bernoulli polynomials of the second kind">Bernoulli polynomials of the second kind</a></li> <li><a href="/wiki/Bernoulli_umbra" title="Bernoulli umbra">Bernoulli umbra</a></li> <li><a href="/wiki/Bell_number" title="Bell number">Bell number</a></li> <li><a href="/wiki/Euler_number" class="mw-redirect" title="Euler number">Euler number</a></li> <li><a href="/wiki/Genocchi_number" title="Genocchi number">Genocchi number</a></li> <li><a href="/wiki/Kummer%27s_congruences" class="mw-redirect" title="Kummer's congruences">Kummer's congruences</a></li> <li><a href="/wiki/Poly-Bernoulli_number" title="Poly-Bernoulli number">Poly-Bernoulli number</a></li> <li><a href="/wiki/Hurwitz_zeta_function" title="Hurwitz zeta function">Hurwitz zeta function</a></li> <li><a href="/wiki/Euler_summation" title="Euler summation">Euler summation</a></li> <li><a href="/wiki/Stirling_polynomial" class="mw-redirect" title="Stirling polynomial">Stirling polynomial</a></li> <li><a href="/wiki/Sums_of_powers" title="Sums of powers">Sums of powers</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=42" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Translation of the text: " ... And if [one were] to proceed onward step by step to higher powers, one may furnish, with little difficulty, the following list: Sums of powers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int n=\sum _{k=1}^{n}k={\frac {1}{2}}n^{2}+{\frac {1}{2}}n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <mi>n</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int n=\sum _{k=1}^{n}k={\frac {1}{2}}n^{2}+{\frac {1}{2}}n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7bbb99311629a3eb9569aeb22e0fe677259148d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.638ex; height:3.509ex;" alt="{\displaystyle \textstyle \int n=\sum _{k=1}^{n}k={\frac {1}{2}}n^{2}+{\frac {1}{2}}n}"> <dl><dd><dl><dd><dl><dd><dl><dd>⋮</dd></dl></dd></dl></dd></dl></dd></dl> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int n^{10}=\sum _{k=1}^{n}k^{10}={\frac {1}{11}}n^{11}+{\frac {1}{2}}n^{10}+{\frac {5}{6}}n^{9}-1n^{7}+1n^{5}-{\frac {1}{2}}n^{3}+{\frac {5}{66}}n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>11</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>66</mn> </mfrac> </mrow> <mi>n</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int n^{10}=\sum _{k=1}^{n}k^{10}={\frac {1}{11}}n^{11}+{\frac {1}{2}}n^{10}+{\frac {5}{6}}n^{9}-1n^{7}+1n^{5}-{\frac {1}{2}}n^{3}+{\frac {5}{66}}n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4df04ef77a53a59b850655ff5784ab0a8e3facfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:67.425ex; height:3.676ex;" alt="{\displaystyle \textstyle \int n^{10}=\sum _{k=1}^{n}k^{10}={\frac {1}{11}}n^{11}+{\frac {1}{2}}n^{10}+{\frac {5}{6}}n^{9}-1n^{7}+1n^{5}-{\frac {1}{2}}n^{3}+{\frac {5}{66}}n}"> Indeed [if] one will have examined diligently the law of arithmetic progression there, one will also be able to continue the same without these circuitous computations: For [if] <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>c</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/628484f3d9a8dceb9fb15c049ab7fb95c8b522ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle \textstyle c}"> is taken as the exponent of any power, the sum of all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle n^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle n^{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6565035e98d23c3489ce1f2805ff391b62053d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.339ex; height:2.176ex;" alt="{\displaystyle \textstyle n^{c}}"> is produced or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int n^{c}=\sum _{k=1}^{n}k^{c}={\frac {1}{c+1}}n^{c+1}+{\frac {1}{2}}n^{c}+{\frac {c}{2}}An^{c-1}+{\frac {c(c-1)(c-2)}{2\cdot 3\cdot 4}}Bn^{c-3}+{\frac {c(c-1)(c-2)(c-3)(c-4)}{2\cdot 3\cdot 4\cdot 5\cdot 6}}Cn^{c-5}+{\frac {c(c-1)(c-2)(c-3)(c-4)(c-5)(c-6)}{2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8}}Dn^{c-7}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>c</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mn>2</mn> </mfrac> </mrow> <mi>A</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mi>B</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> <mo>⋅</mo> <mn>5</mn> <mo>⋅</mo> <mn>6</mn> </mrow> </mfrac> </mrow> <mi>C</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>−</mo> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> <mo>⋅</mo> <mn>5</mn> <mo>⋅</mo> <mn>6</mn> <mo>⋅</mo> <mn>7</mn> <mo>⋅</mo> <mn>8</mn> </mrow> </mfrac> </mrow> <mi>D</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>−</mo> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int n^{c}=\sum _{k=1}^{n}k^{c}={\frac {1}{c+1}}n^{c+1}+{\frac {1}{2}}n^{c}+{\frac {c}{2}}An^{c-1}+{\frac {c(c-1)(c-2)}{2\cdot 3\cdot 4}}Bn^{c-3}+{\frac {c(c-1)(c-2)(c-3)(c-4)}{2\cdot 3\cdot 4\cdot 5\cdot 6}}Cn^{c-5}+{\frac {c(c-1)(c-2)(c-3)(c-4)(c-5)(c-6)}{2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8}}Dn^{c-7}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d044bc348358685999270bdba35c0ebcc270b30f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:130.737ex; height:4.343ex;" alt="{\displaystyle \textstyle \int n^{c}=\sum _{k=1}^{n}k^{c}={\frac {1}{c+1}}n^{c+1}+{\frac {1}{2}}n^{c}+{\frac {c}{2}}An^{c-1}+{\frac {c(c-1)(c-2)}{2\cdot 3\cdot 4}}Bn^{c-3}+{\frac {c(c-1)(c-2)(c-3)(c-4)}{2\cdot 3\cdot 4\cdot 5\cdot 6}}Cn^{c-5}+{\frac {c(c-1)(c-2)(c-3)(c-4)(c-5)(c-6)}{2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8}}Dn^{c-7}+\cdots }"> and so forth, the exponent of its power <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> continually diminishing by 2 until it arrives at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{2}}">. The capital letters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle A,B,C,D,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> <mo>,</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle A,B,C,D,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/897ff036841616540e558f9c8b1e22d21ca97dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.946ex; height:2.509ex;" alt="{\displaystyle \textstyle A,B,C,D,}"> etc. denote in order the coefficients of the last terms for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int n^{2},\int n^{4},\int n^{6},\int n^{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo>∫</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>,</mo> <mo>∫</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>,</mo> <mo>∫</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int n^{2},\int n^{4},\int n^{6},\int n^{8}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c06697bbe7c8cf6dc0a5c79727f751e4461d85d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.118ex; height:3.176ex;" alt="{\displaystyle \textstyle \int n^{2},\int n^{4},\int n^{6},\int n^{8}}">, etc. namely <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle A={\frac {1}{6}},B=-{\frac {1}{30}},C={\frac {1}{42}},D=-{\frac {1}{30}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>,</mo> <mi>B</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> <mo>,</mo> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>42</mn> </mfrac> </mrow> <mo>,</mo> <mi>D</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle A={\frac {1}{6}},B=-{\frac {1}{30}},C={\frac {1}{42}},D=-{\frac {1}{30}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73098a2164694ccaebddc38dc9d933edf9898888" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:35.408ex; height:3.676ex;" alt="{\displaystyle \textstyle A={\frac {1}{6}},B=-{\frac {1}{30}},C={\frac {1}{42}},D=-{\frac {1}{30}}}">." [Note: The text of the illustration contains some typos: ensperexit should read inspexerit, ambabimus should read ambagibus, quosque should read quousque, and in Bernoulli's original text Sumtâ should read Sumptâ or Sumptam.] <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSmith1929" class="citation cs2">Smith, David Eugene (1929), <a rel="nofollow" class="external text" href="https://archive.org/details/sourcebookinmath00smit/page/85">"Jacques (I) Bernoulli: On the 'Bernoulli Numbers'"</a>, A Source Book in Mathematics, New York: McGraw-Hill Book Co., pp. 85–90</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Jacques+%28I%29+Bernoulli%3A+On+the+%27Bernoulli+Numbers%27&rft.btitle=A+Source+Book+in+Mathematics&rft.place=New+York&rft.pages=85-90&rft.pub=McGraw-Hill+Book+Co.&rft.date=1929&rft.aulast=Smith&rft.aufirst=David+Eugene&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsourcebookinmath00smit%2Fpage%2F85&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernoulli1713" class="citation cs2 cs1-prop-foreign-lang-source">Bernoulli, Jacob (1713), <a rel="nofollow" class="external text" href="https://archive.org/details/jacobibernoulli00bern/page/97">Ars Conjectandi</a> (in Latin), Basel: Impensis Thurnisiorum, Fratrum, pp. 97–98, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5479%2Fsil.262971.39088000323931">10.5479/sil.262971.39088000323931</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ars+Conjectandi&rft.place=Basel&rft.pages=97-98&rft.pub=Impensis+Thurnisiorum%2C+Fratrum&rft.date=1713&rft_id=info%3Adoi%2F10.5479%2Fsil.262971.39088000323931&rft.aulast=Bernoulli&rft.aufirst=Jacob&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fjacobibernoulli00bern%2Fpage%2F97&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"></li></ul> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> The <a href="#CITEREFMathematics_Genealogy_Projectn.d.">Mathematics Genealogy Project (n.d.)</a> shows Leibniz as the academic advisor of Jakob Bernoulli. See also <a href="#CITEREFMiller2017">Miller (2017)</a>. </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> this formula was discovered (or perhaps rediscovered) by Giorgio Pietrocola. His demonstration is available in Italian language (<a href="#CITEREFPietrocola2008">Pietrocola 2008</a>). </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=43" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Weisstein2016-1">^ <a href="#cite_ref-Weisstein2016_1-0">a</a> <a href="#cite_ref-Weisstein2016_1-1">b</a> <a href="#cite_ref-Weisstein2016_1-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs2"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/BernoulliNumber.html">"Bernoulli Number"</a>, <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Bernoulli+Number&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FBernoulliNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"> </li> <li id="cite_note-Selin1997_891-2">^ <a href="#cite_ref-Selin1997_891_2-0">a</a> <a href="#cite_ref-Selin1997_891_2-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSelin1997" class="citation cs2"><a href="/wiki/Helaine_Selin" title="Helaine Selin">Selin, Helaine</a>, ed. (1997), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, p. 819 (p. 891), <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008ehst.book.....S">2008ehst.book.....S</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-4066-3" title="Special:BookSources/0-7923-4066-3"><bdi>0-7923-4066-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopaedia+of+the+History+of+Science%2C+Technology%2C+and+Medicine+in+Non-Western+Cultures&rft.pages=p.+819+%28p.+891%29&rft.pub=Springer&rft.date=1997&rft_id=info%3Abibcode%2F2008ehst.book.....S&rft.isbn=0-7923-4066-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"> </li> <li id="cite_note-SmithMikami1914_108-3"><a href="#cite_ref-SmithMikami1914_108_3-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmithMikami1914" class="citation cs2">Smith, David Eugene; Mikami, Yoshio (1914), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pTcQsvfbSu4C">A history of Japanese mathematics</a>, Open Court publishing company, p. 108, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486434827" title="Special:BookSources/9780486434827"><bdi>9780486434827</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+Japanese+mathematics&rft.pages=108&rft.pub=Open+Court+publishing+company&rft.date=1914&rft.isbn=9780486434827&rft.aulast=Smith&rft.aufirst=David+Eugene&rft.au=Mikami%2C+Yoshio&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpTcQsvfbSu4C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"> </li> <li id="cite_note-Kitagawa-4"><a href="#cite_ref-Kitagawa_4-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKitagawa2021" class="citation cs2">Kitagawa, Tomoko L. (2021-07-23), "The Origin of the Bernoulli Numbers: Mathematics in Basel and Edo in the Early Eighteenth Century", The Mathematical Intelligencer, 44: 46–56, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00283-021-10072-y">10.1007/s00283-021-10072-y</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0343-6993">0343-6993</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Intelligencer&rft.atitle=The+Origin+of+the+Bernoulli+Numbers%3A+Mathematics+in+Basel+and+Edo+in+the+Early+Eighteenth+Century&rft.volume=44&rft.pages=46-56&rft.date=2021-07-23&rft_id=info%3Adoi%2F10.1007%2Fs00283-021-10072-y&rft.issn=0343-6993&rft.aulast=Kitagawa&rft.aufirst=Tomoko+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"> </li> <li id="cite_note-Menabrea1842_noteG-5"><a href="#cite_ref-Menabrea1842_noteG_5-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMenabrea1842" class="citation cs2">Menabrea, L.F. (1842), <a rel="nofollow" class="external text" href="http://www.fourmilab.ch/babbage/sketch.html">"Sketch of the Analytic Engine invented by Charles Babbage, with notes upon the Memoir by the Translator Ada Augusta, Countess of Lovelace"</a>, Bibliothèque Universelle de Genève, 82, See Note G</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biblioth%C3%A8que+Universelle+de+Gen%C3%A8ve&rft.atitle=Sketch+of+the+Analytic+Engine+invented+by+Charles+Babbage%2C+with+notes+upon+the+Memoir+by+the+Translator+Ada+Augusta%2C+Countess+of+Lovelace&rft.volume=82&rft.pages=See+%27%27Note+G%27%27&rft.date=1842&rft.aulast=Menabrea&rft.aufirst=L.F.&rft_id=http%3A%2F%2Fwww.fourmilab.ch%2Fbabbage%2Fsketch.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArfken1970278-6"><a href="#cite_ref-FOOTNOTEArfken1970278_6-0">^</a> <a href="#CITEREFArfken1970">Arfken (1970)</a>, p. 278. </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a> (2022), <a rel="nofollow" class="external text" href="https://www-cs-faculty.stanford.edu/~knuth/news22.html">Recent News (2022): Concrete Mathematics and Bernoulli</a>. <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">But last year I took a close look at Peter Luschny's Bernoulli manifesto, where he gives more than a dozen good reasons why the value of $B_1$ should really be plus one-half. He explains that some mathematicians of the early 20th century had unilaterally changed the conventions, because some of their formulas came out a bit nicer when the negative value was used. It was their well-intentioned but ultimately poor choice that had led to what I'd been taught in the 1950s. […] By now, hundreds of books that use the “minus-one-half” convention have unfortunately been written. Even worse, all the major software systems for symbolic mathematics have that 20th-century aberration deeply embedded. Yet Luschny convinced me that we have all been wrong, and that it's high time to change back to the correct definition before the situation gets even worse. </blockquote> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> Peter Luschny (2013), <a rel="nofollow" class="external text" href="http://luschny.de/math/zeta/The-Bernoulli-Manifesto.html">The Bernoulli Manifesto</a> </li> <li id="cite_note-FOOTNOTEKnuth1993-9">^ <a href="#cite_ref-FOOTNOTEKnuth1993_9-0">a</a> <a href="#cite_ref-FOOTNOTEKnuth1993_9-1">b</a> <a href="#CITEREFKnuth1993">Knuth (1993)</a>. </li> <li id="cite_note-Jacobi1834-10"><a href="#cite_ref-Jacobi1834_10-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacobi1834" class="citation cs2"><a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Jacobi, C.G.J.</a> (1834), <a rel="nofollow" class="external text" href="https://zenodo.org/record/1448824">"De usu legitimo formulae summatoriae Maclaurinianae"</a>, Journal für die reine und angewandte Mathematik, 12: 263–272</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+reine+und+angewandte+Mathematik&rft.atitle=De+usu+legitimo+formulae+summatoriae+Maclaurinianae&rft.volume=12&rft.pages=263-272&rft.date=1834&rft.aulast=Jacobi&rft.aufirst=C.G.J.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1448824&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"> </li> <li id="cite_note-FOOTNOTEKnuth199314-11"><a href="#cite_ref-FOOTNOTEKnuth199314_11-0">^</a> <a href="#CITEREFKnuth1993">Knuth (1993)</a>, p. 14. </li> <li id="cite_note-FOOTNOTEGrahamKnuthPatashnik1989Section_2.51-14"><a href="#cite_ref-FOOTNOTEGrahamKnuthPatashnik1989Section_2.51_14-0">^</a> <a href="#CITEREFGrahamKnuthPatashnik1989">Graham, Knuth & Patashnik (1989)</a>, Section 2.51. </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> See <a href="#CITEREFIrelandRosen1990">Ireland & Rosen (1990)</a> or <a href="#CITEREFConwayGuy1996">Conway & Guy (1996)</a>. </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> Jordan (1950) p 233 </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> Ireland and Rosen (1990) p 229 </li> <li id="cite_note-Saalschütz1893-18"><a href="#cite_ref-Saalschütz1893_18-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSaalschütz1893" class="citation cs2">Saalschütz, Louis (1893), <a rel="nofollow" class="external text" href="http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=00450002">Vorlesungen über die Bernoullischen Zahlen, ihren Zusammenhang mit den Secanten-Coefficienten und ihre wichtigeren Anwendungen</a>, Berlin: Julius Springer</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vorlesungen+%C3%BCber+die+Bernoullischen+Zahlen%2C+ihren+Zusammenhang+mit+den+Secanten-Coefficienten+und+ihre+wichtigeren+Anwendungen&rft.place=Berlin&rft.pub=Julius+Springer&rft.date=1893&rft.aulast=Saalsch%C3%BCtz&rft.aufirst=Louis&rft_id=http%3A%2F%2Fdigital.library.cornell.edu%2Fcgi%2Ft%2Ftext%2Ftext-idx%3Fc%3Dmath%3Bidno%3D00450002&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988">. </li> <li id="cite_note-FOOTNOTEArfken1970279-19"><a href="#cite_ref-FOOTNOTEArfken1970279_19-0">^</a> <a href="#CITEREFArfken1970">Arfken (1970)</a>, p. 279. </li> <li id="cite_note-BuhlerCraErnMetShokrollahi2001-20"><a href="#cite_ref-BuhlerCraErnMetShokrollahi2001_20-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuhlerCrandallErnvallMetsankyla2001" class="citation cs2">Buhler, J.; Crandall, R.; Ernvall, R.; Metsankyla, T.; Shokrollahi, M. (2001), "Irregular Primes and Cyclotomic Invariants to 12 Million", Journal of Symbolic Computation, 31 (1–2): 89–96, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjsco.1999.1011">10.1006/jsco.1999.1011</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Computation&rft.atitle=Irregular+Primes+and+Cyclotomic+Invariants+to+12+Million&rft.volume=31&rft.issue=1%E2%80%932&rft.pages=89-96&rft.date=2001&rft_id=info%3Adoi%2F10.1006%2Fjsco.1999.1011&rft.aulast=Buhler&rft.aufirst=J.&rft.au=Crandall%2C+R.&rft.au=Ernvall%2C+R.&rft.au=Metsankyla%2C+T.&rft.au=Shokrollahi%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"> </li> <li id="cite_note-Harvey2010-21"><a href="#cite_ref-Harvey2010_21-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarvey2010" class="citation cs2">Harvey, David (2010), "A multimodular algorithm for computing Bernoulli numbers", Math. Comput., 79 (272): 2361–2370, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0807.1347">0807.1347</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-2010-02367-1">10.1090/S0025-5718-2010-02367-1</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11329343">11329343</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1215.11016">1215.11016</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math.+Comput.&rft.atitle=A+multimodular+algorithm+for+computing+Bernoulli+numbers&rft.volume=79&rft.issue=272&rft.pages=2361-2370&rft.date=2010&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1215.11016%23id-name%3DZbl&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11329343%23id-name%3DS2CID&rft_id=info%3Aarxiv%2F0807.1347&rft_id=info%3Adoi%2F10.1090%2FS0025-5718-2010-02367-1&rft.aulast=Harvey&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"> </li> <li id="cite_note-Kellner2002-22"><a href="#cite_ref-Kellner2002_22-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKellner2002" class="citation cs2">Kellner, Bernd (2002), <a rel="nofollow" class="external text" href="http://www.bernoulli.org/">Program Calcbn – A program for calculating Bernoulli numbers</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Program+Calcbn+%E2%80%93+A+program+for+calculating+Bernoulli+numbers&rft.date=2002&rft.aulast=Kellner&rft.aufirst=Bernd&rft_id=http%3A%2F%2Fwww.bernoulli.org%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988">. </li> <li id="cite_note-Pavlyk29Apr2008-23"><a href="#cite_ref-Pavlyk29Apr2008_23-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPavlyk2008" class="citation cs2">Pavlyk, Oleksandr (29 April 2008), <a rel="nofollow" class="external text" href="http://blog.wolfram.com/2008/04/29/today-we-broke-the-bernoulli-record-from-the-analytical-engine-to-mathematica/">"Today We Broke the Bernoulli Record: From the Analytical Engine to Mathematica"</a>, Wolfram News</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Wolfram+News&rft.atitle=Today+We+Broke+the+Bernoulli+Record%3A+From+the+Analytical+Engine+to+Mathematica&rft.date=2008-04-29&rft.aulast=Pavlyk&rft.aufirst=Oleksandr&rft_id=http%3A%2F%2Fblog.wolfram.com%2F2008%2F04%2F29%2Ftoday-we-broke-the-bernoulli-record-from-the-analytical-engine-to-mathematica%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988">. </li> <li id="cite_note-FOOTNOTEGrahamKnuthPatashnik19899.67-24"><a href="#cite_ref-FOOTNOTEGrahamKnuthPatashnik19899.67_24-0">^</a> <a href="#CITEREFGrahamKnuthPatashnik1989">Graham, Knuth & Patashnik (1989)</a>, 9.67. </li> <li id="cite_note-FOOTNOTEGrahamKnuthPatashnik19892.44,_2.52-25"><a href="#cite_ref-FOOTNOTEGrahamKnuthPatashnik19892.44,_2.52_25-0">^</a> <a href="#CITEREFGrahamKnuthPatashnik1989">Graham, Knuth & Patashnik (1989)</a>, 2.44, 2.52. </li> <li id="cite_note-GuoZeng2005-26"><a href="#cite_ref-GuoZeng2005_26-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuoZeng2005" class="citation cs2">Guo, Victor J. W.; Zeng, Jiang (30 August 2005), "A q-Analogue of Faulhaber's Formula for Sums of Powers", The Electronic Journal of Combinatorics, 11 (2), <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0501441">math/0501441</a>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005math......1441G">2005math......1441G</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.37236%2F1876">10.37236/1876</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10467873">10467873</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Electronic+Journal+of+Combinatorics&rft.atitle=A+q-Analogue+of+Faulhaber%27s+Formula+for+Sums+of+Powers&rft.volume=11&rft.issue=2&rft.date=2005-08-30&rft_id=info%3Aarxiv%2Fmath%2F0501441&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10467873%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.37236%2F1876&rft_id=info%3Abibcode%2F2005math......1441G&rft.aulast=Guo&rft.aufirst=Victor+J.+W.&rft.au=Zeng%2C+Jiang&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"> </li> <li id="cite_note-FOOTNOTEArfken1970463-27"><a href="#cite_ref-FOOTNOTEArfken1970463_27-0">^</a> <a href="#CITEREFArfken1970">Arfken (1970)</a>, p. 463. </li> <li id="cite_note-Rademacher1973-28">^ <a href="#cite_ref-Rademacher1973_28-0">a</a> <a href="#cite_ref-Rademacher1973_28-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRademacher1973" class="citation cs2">Rademacher, H. 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(1883), <a rel="nofollow" class="external text" href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002158698">"Studien über die Bernoullischen und Eulerschen Zahlen"</a>, Journal für die reine und angewandte Mathematik, 94: 203–232</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+reine+und+angewandte+Mathematik&rft.atitle=Studien+%C3%BCber+die+Bernoullischen+und+Eulerschen+Zahlen&rft.volume=94&rft.pages=203-232&rft.date=1883&rft.aulast=Worpitzky&rft.aufirst=J.&rft_id=http%3A%2F%2Fresolver.sub.uni-goettingen.de%2Fpurl%3FGDZPPN002158698&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988">.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Bernoulli_number&action=edit&section=45" title="Edit section: External links">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Bernoulli_numbers">"Bernoulli numbers"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bernoulli+numbers&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DBernoulli_numbers&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"></li> <li><a href="https://www.gutenberg.org/ebooks/2586" class="extiw" title="gutenberg:2586">The first 498 Bernoulli Numbers</a> from <a href="/wiki/Project_Gutenberg" title="Project Gutenberg">Project Gutenberg</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304004018/http://web.maths.unsw.edu.au/~davidharvey/papers/bernmm/">A multimodular algorithm for computing Bernoulli numbers</a></li> <li><a rel="nofollow" class="external text" href="http://www.bernoulli.org">The Bernoulli Number Page</a></li> <li><a href="/w/index.php?title=Literateprograms:Category:Bernoulli_numbers&action=edit&redlink=1" class="new" title="Literateprograms:Category:Bernoulli numbers (page does not exist)">Bernoulli number programs</a> at <a rel="nofollow" class="external text" href="http://en.literateprograms.org">LiteratePrograms</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFP._Luschny" class="citation cs2">P. Luschny, <a rel="nofollow" class="external text" href="http://www.luschny.de/math/primes/irregular.html">The Computation of Irregular Primes</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Computation+of+Irregular+Primes&rft.au=P.+Luschny&rft_id=http%3A%2F%2Fwww.luschny.de%2Fmath%2Fprimes%2Firregular.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFP._Luschny" class="citation cs2">P. Luschny, <a rel="nofollow" class="external text" href="http://oeis.org/wiki/User:Peter_Luschny/ComputationAndAsymptoticsOfBernoulliNumbers">The Computation And Asymptotics Of Bernoulli Numbers</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Computation+And+Asymptotics+Of+Bernoulli+Numbers&rft.au=P.+Luschny&rft_id=http%3A%2F%2Foeis.org%2Fwiki%2FUser%3APeter_Luschny%2FComputationAndAsymptoticsOfBernoulliNumbers&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGottfried_Helms" class="citation cs2">Gottfried Helms, <a rel="nofollow" class="external text" href="http://go.helms-net.de/math/pascal/bernoulli_en.pdf">Bernoullinumbers in context of Pascal-(Binomial)matrix</a> (PDF), <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://go.helms-net.de/math/pascal/bernoulli_en.pdf">archived</a> (PDF) from the original on 2022-10-09</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bernoullinumbers+in+context+of+Pascal-%28Binomial%29matrix&rft.au=Gottfried+Helms&rft_id=http%3A%2F%2Fgo.helms-net.de%2Fmath%2Fpascal%2Fbernoulli_en.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGottfried_Helms" class="citation cs2">Gottfried Helms, <a rel="nofollow" class="external text" href="http://go.helms-net.de/math/binomial/04_3_SummingOfLikePowers.pdf">summing of like powers in context with Pascal-/Bernoulli-matrix</a> (PDF), <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://go.helms-net.de/math/binomial/04_3_SummingOfLikePowers.pdf">archived</a> (PDF) from the original on 2022-10-09</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=summing+of+like+powers+in+context+with+Pascal-%2FBernoulli-matrix&rft.au=Gottfried+Helms&rft_id=http%3A%2F%2Fgo.helms-net.de%2Fmath%2Fbinomial%2F04_3_SummingOfLikePowers.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGottfried_Helms" class="citation cs2">Gottfried Helms, <a rel="nofollow" class="external text" href="http://go.helms-net.de/math/binomial/02_2_GeneralizedBernoulliRecursion.pdf">Some special properties, sums of Bernoulli-and related numbers</a> (PDF), <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://go.helms-net.de/math/binomial/02_2_GeneralizedBernoulliRecursion.pdf">archived</a> (PDF) from the original on 2022-10-09</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Some+special+properties%2C+sums+of+Bernoulli-and+related+numbers&rft.au=Gottfried+Helms&rft_id=http%3A%2F%2Fgo.helms-net.de%2Fmath%2Fbinomial%2F02_2_GeneralizedBernoulliRecursion.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABernoulli+number" class="Z3988"></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline 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href="/wiki/Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">Inverse functions and differentiation</a></li> <li><a href="/wiki/Logarithmic_derivative" title="Logarithmic derivative">Logarithmic derivative</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li></ul></li> <li><a href="/wiki/Stationary_point" title="Stationary point">Stationary points</a> <ul><li><a href="/wiki/First_derivative_test" class="mw-redirect" title="First derivative test">First derivative test</a></li> <li><a href="/wiki/Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a></li> <li><a href="/wiki/Extreme_value_theorem" title="Extreme value theorem">Extreme value theorem</a></li> <li><a href="/wiki/Maximum_and_minimum" title="Maximum and minimum">Maximum and minimum</a></li></ul></li> <li>Further applications <ul><li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equation</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">Integral calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Arc_length" title="Arc length">Arc length</a></li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Integral#Properties" title="Integral">Basic properties</a></li> <li><a href="/wiki/Constant_of_integration" title="Constant of integration">Constant of integration</a></li> <li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a> <ul><li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Differentiating under the integral sign</a></li></ul></li> <li><a href="/wiki/Integration_by_parts" title="Integration by parts">Integration by parts</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Integration by substitution</a> <ul><li><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a></li> <li><a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a></li> <li><a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">Tangent half-angle substitution</a></li></ul></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a> <ul><li><a href="/wiki/Quadratic_integral" title="Quadratic integral">Quadratic integral</a></li></ul></li> <li><a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li> <li>Volumes <ul><li><a href="/wiki/Disc_integration" title="Disc integration">Washer method</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Shell method</a></li></ul></li> <li><a href="/wiki/Integral_equation" title="Integral equation">Integral equation</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Derivatives <ul><li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li></ul></li> <li>Basic theorems <ul><li><a href="/wiki/Fundamental_Theorem_of_Line_Integrals" class="mw-redirect" title="Fundamental Theorem of Line Integrals">Line integrals</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Gauss'</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix and determinant</a></li> <li><a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li>Advanced topics <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Generalized_Stokes%27_theorem" class="mw-redirect" title="Generalized Stokes' theorem">Generalized Stokes' theorem</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sequences and series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetico-geometric_sequence" title="Arithmetico-geometric sequence">Arithmetico-geometric sequence</a></li> <li>Types of series <ul><li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier</a></li> <li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">Infinite</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a> <ul><li><a href="/wiki/Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul></li> <li>Tests of convergence <ul><li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel's</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet's</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Term</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special functions and numbers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Bernoulli numbers</a></li> <li><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></li> <li><a href="/wiki/Exponential_function" title="Exponential function">Exponential function</a></li> <li><a href="/wiki/Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li> <li><a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_calculus" title="History of calculus">History of calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adequality" title="Adequality">Adequality</a></li> <li><a href="/wiki/Brook_Taylor" title="Brook Taylor">Brook Taylor</a></li> <li><a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a></li> <li><a href="/wiki/Generality_of_algebra" title="Generality of algebra">Generality of algebra</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimal</a></li> <li><a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">Infinitesimal calculus</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Fluxion" title="Fluxion">Fluxion</a></li> <li><a href="/wiki/Law_of_Continuity" class="mw-redirect" title="Law of Continuity">Law of Continuity</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></li> <li><a href="/wiki/The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Integrals" scope="row" class="navbox-group" style="width:1%;text-align:left"><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_integrals_of_rational_functions" title="List of integrals of rational functions">rational functions</a></li> <li><a href="/wiki/List_of_integrals_of_irrational_functions" title="List of integrals of irrational functions">irrational functions</a></li> <li><a href="/wiki/List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">exponential functions</a></li> <li><a href="/wiki/List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">logarithmic functions</a></li> <li><a href="/wiki/List_of_integrals_of_hyperbolic_functions" title="List of integrals of hyperbolic functions">hyperbolic functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_hyperbolic_functions" title="List of integrals of inverse hyperbolic functions">inverse</a></li></ul></li> <li><a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">trigonometric functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse</a></li> <li><a href="/wiki/Integral_of_the_secant_function" title="Integral of the secant function">Secant</a></li> <li><a href="/wiki/Integral_of_secant_cubed" title="Integral of secant cubed">Secant cubed</a></li></ul></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_limits" title="List of limits">List of limits</a></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">List of derivatives</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Complex calculus <ul><li><a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a></li></ul></li> <li>Differential geometry <ul><li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Curvature" title="Curvature">Curvature</a></li> <li><a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">of curves</a></li> <li><a href="/wiki/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">of surfaces</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li></ul></li> <li><a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li> <li><a href="/wiki/Gabriel%27s_horn" title="Gabriel's horn">Gabriel's horn</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds 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7.64% 176.446 43 Template:R/ref"," 6.17% 142.487 1 Template:Short_description"," 5.98% 138.246 1 Template:Neukirch_ANT"," 5.88% 135.777 1 Template:Cite_book"]},"scribunto":{"limitreport-timeusage":{"value":"1.206","limit":"10.000"},"limitreport-memusage":{"value":21969358,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAbramowitzStegun1972\"] = 1,\n [\"CITEREFAgohDilcher2008\"] = 1,\n [\"CITEREFAndré1879\"] = 1,\n [\"CITEREFAndré1881\"] = 1,\n [\"CITEREFApostol2010\"] = 1,\n [\"CITEREFArfken1970\"] = 1,\n [\"CITEREFArlettaz1998\"] = 1,\n [\"CITEREFArnold1991\"] = 1,\n [\"CITEREFAyoub1981\"] = 1,\n [\"CITEREFBannaiKobayashi2010\"] = 1,\n [\"CITEREFBernoulli1713\"] = 1,\n [\"CITEREFBoole1880\"] = 1,\n [\"CITEREFBuhlerCrandallErnvallMetsankyla2001\"] = 1,\n [\"CITEREFCarlitz1968\"] = 1,\n [\"CITEREFCharolloisSczech2016\"] = 1,\n [\"CITEREFClausen1840\"] = 1,\n [\"CITEREFConwayGuy1996\"] = 1,\n [\"CITEREFDilcherSkulaSlavutskii1991\"] = 1,\n [\"CITEREFDumont1981\"] = 1,\n [\"CITEREFDumontViennot1980\"] = 1,\n [\"CITEREFElkies2003\"] = 1,\n [\"CITEREFEntringer1966\"] = 1,\n [\"CITEREFEuler1735\"] = 1,\n [\"CITEREFFeePlouffe2007\"] = 1,\n [\"CITEREFGottfried_Helms\"] = 3,\n [\"CITEREFGould1972\"] = 1,\n [\"CITEREFGrahamKnuthPatashnik1989\"] = 1,\n [\"CITEREFGuoZeng2005\"] = 1,\n [\"CITEREFHarvey2010\"] = 1,\n [\"CITEREFIrelandRosen1990\"] = 1,\n [\"CITEREFJacobi1834\"] = 1,\n [\"CITEREFJordan1950\"] = 1,\n [\"CITEREFKaneko2000\"] = 1,\n [\"CITEREFKellner2002\"] = 1,\n [\"CITEREFKitagawa2021\"] = 1,\n [\"CITEREFKnuth1993\"] = 1,\n [\"CITEREFKnuthBuckholtz1967\"] = 1,\n [\"CITEREFKummer1850\"] = 1,\n [\"CITEREFKummer1851\"] = 1,\n [\"CITEREFLuschny2007\"] = 1,\n [\"CITEREFLuschny2011\"] = 1,\n [\"CITEREFMalenfant2011\"] = 1,\n [\"CITEREFMathematics_Genealogy_Projectn.d.\"] = 1,\n [\"CITEREFMenabrea1842\"] = 1,\n [\"CITEREFMiller2017\"] = 1,\n [\"CITEREFMilnorStasheff1974\"] = 1,\n [\"CITEREFP._Luschny\"] = 2,\n [\"CITEREFPavlyk2008\"] = 1,\n [\"CITEREFPietrocola2008\"] = 1,\n [\"CITEREFRademacher1973\"] = 1,\n [\"CITEREFRiesz1916\"] = 1,\n [\"CITEREFSaalschütz1893\"] = 1,\n [\"CITEREFSeidel1877\"] = 1,\n [\"CITEREFSelin1997\"] = 1,\n [\"CITEREFSlavutskii1995\"] = 1,\n [\"CITEREFSmith1929\"] = 1,\n [\"CITEREFSmithMikami1914\"] = 1,\n [\"CITEREFStanley2010\"] = 1,\n [\"CITEREFSun2005–2006\"] = 1,\n [\"CITEREFWoon1997\"] = 1,\n [\"CITEREFWoon1998\"] = 1,\n [\"CITEREFWorpitzky1883\"] = 1,\n [\"CITEREFvon_Ettingshausen1827\"] = 1,\n [\"CITEREFvon_Staudt1840\"] = 1,\n [\"CITEREFvon_Staudt1845\"] = 1,\n}\ntemplate_list = table#1 {\n [\"\"] = 1,\n [\"1\\\\over x\"] = 2,\n [\"=\"] = 54,\n [\"Abs\"] = 1,\n [\"Authority control\"] = 1,\n [\"Blockquote\"] = 1,\n [\"Calculus topics\"] = 1,\n [\"Citation\"] = 65,\n [\"Cite arXiv\"] = 3,\n [\"Collapse bottom\"] = 1,\n [\"Collapse top\"] = 1,\n [\"Diagonal split header\"] = 2,\n [\"Div col\"] = 1,\n [\"Div col end\"] = 1,\n [\"Efn\"] = 3,\n [\"Harv\"] = 1,\n [\"Harvp\"] = 4,\n [\"Image frame\"] = 1,\n [\"Main\"] = 5,\n [\"Math\"] = 316,\n [\"Mathworld\"] = 1,\n [\"Mvar\"] = 58,\n [\"N+1\\\\atop m+1\"] = 1,\n [\"Neukirch ANT\"] = 1,\n [\"Notelist\"] = 1,\n [\"Nowrap\"] = 8,\n [\"OEIS2C\"] = 94,\n [\"Overline\"] = 2,\n [\"Pi\"] = 6,\n [\"R\"] = 37,\n [\"Reflist\"] = 1,\n [\"SfnRef\"] = 1,\n [\"Sfnp\"] = 9,\n [\"Sfrac\"] = 271,\n [\"Short description\"] = 1,\n [\"SpringerEOM\"] = 1,\n [\"Su\"] = 14,\n [\"Sub\"] = 3,\n [\"Sup\"] = 1,\n [\"Table alignment\"] = 1,\n [\"Underline\"] = 2,\n [\"Unordered list\"] = 1,\n [\"Use American English\"] = 1,\n [\"Use shortened footnotes\"] = 1,\n [\"Val\"] = 12,\n}\narticle_whitelist = table#1 {\n}\n","limitreport-profile":[["?","260","18.3"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::getAllExpandedArguments","240","16.9"],["recursiveClone 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number","url":"https:\/\/en.wikipedia.org\/wiki\/Bernoulli_number","sameAs":"http:\/\/www.wikidata.org\/entity\/Q694114","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q694114","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-01-11T23:37:01Z","dateModified":"2024-11-23T02:24:14Z","headline":"rational number sequence \ud835\udc35\u2096 such that (\ud835\udc5a+1)\u2211\ud835\udc5b\u1d50=(\u1d50\u207a\u00b9\u2080)\ud835\udc35\u2080\ud835\udc5b\u1d50\u207a\u00b9\u2212(\u1d50\u207a\u00b9\u2081)\ud835\udc35\u2081\ud835\udc5b\u1d50+(\u1d50\u207a\u00b9\u2082)\ud835\udc35\u2082\ud835\udc5b\u1d50\u00af\u00b9\u2212(\u1d50\u207a\u00b9\u2083)\ud835\udc35\u2083\ud835\udc5b\u1d50\u00af\u00b2+\u22ef"}</script> </body> </html>

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Bernoulli number - Wikipedia