Faktorialo - Vikipedio

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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Ensaluti</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Paĝoj por elsalutitaj redaktistoj <a href="/wiki/Helpo:Enkonduko" aria-label="Lernu pli pri redaktado"><span>Lernu pli</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Speciala%C4%B5o:Miaj_kontribuoj" title="Listo de redaktoj faritaj el ĉi tiu IPa adreso [y]" accesskey="y"><span>Kontribuoj</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Speciala%C4%B5o:Mia_diskutpa%C4%9Do" title="Diskuto pri redaktoj sub tiu ĉi IP adreso [n]" accesskey="n"><span>Diskuto</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div 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<span class="vector-toc-numb">2</span> <span>Kombinatoriko</span> </div> </a> <ul id="toc-Kombinatoriko-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Γ-funkcio" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Γ-funkcio"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Γ-funkcio</span> </div> </a> <ul id="toc-Γ-funkcio-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proksimuma_kalkulado_de_Stirling" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proksimuma_kalkulado_de_Stirling"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Proksimuma kalkulado de Stirling</span> </div> </a> <ul id="toc-Proksimuma_kalkulado_de_Stirling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Duopa_faktorialo" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Duopa_faktorialo"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Duopa faktorialo</span> </div> </a> <ul id="toc-Duopa_faktorialo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Plurfaktorialo" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Plurfaktorialo"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Plurfaktorialo</span> </div> </a> <ul id="toc-Plurfaktorialo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primofaktorialo" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Primofaktorialo"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Primofaktorialo</span> </div> </a> <ul id="toc-Primofaktorialo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vidu_ankaŭ" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vidu_ankaŭ"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Vidu ankaŭ</span> </div> </a> <ul id="toc-Vidu_ankaŭ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eksteraj_ligiloj" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Eksteraj_ligiloj"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Eksteraj ligiloj</span> </div> </a> <ul id="toc-Eksteraj_ligiloj-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="Enhavo" 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="Ŝaltu la enhavtabelon" > <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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Ŝaltu la enhavtabelon</span> </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"><span class="mw-page-title-main">Faktorialo</span></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" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Iru al artikolo en alia lingvo. Havebla en 80 lingvo" > <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-80" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">80 lingvoj</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Fakulteit_(wiskunde)" title="Fakulteit (wiskunde) — afrikansa" lang="af" hreflang="af" data-title="Fakulteit (wiskunde)" data-language-autonym="Afrikaans" data-language-local-name="afrikansa" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8D%8B%E1%8A%AD%E1%89%B6%E1%88%AA%E1%8B%AB%E1%88%8D" title="ፋክቶሪያል — amhara" lang="am" hreflang="am" data-title="ፋክቶሪያል" data-language-autonym="አማርኛ" data-language-local-name="amhara" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%A7%D9%85%D9%84%D9%8A" title="عاملي — araba" lang="ar" hreflang="ar" data-title="عاملي" data-language-autonym="العربية" data-language-local-name="araba" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Factorial" title="Factorial — asturia" lang="ast" hreflang="ast" data-title="Factorial" data-language-autonym="Asturianu" data-language-local-name="asturia" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Faktorial" title="Faktorial — azerbajĝana" lang="az" hreflang="az" data-title="Faktorial" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajĝana" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B0%D0%BB" title="Факториал — baŝkira" lang="ba" hreflang="ba" data-title="Факториал" data-language-autonym="Башҡортса" data-language-local-name="baŝkira" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%B0%D1%80%D1%8B%D1%8F%D0%BB" title="Фактарыял — belorusa" lang="be" hreflang="be" data-title="Фактарыял" data-language-autonym="Беларуская" data-language-local-name="belorusa" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B5%D0%BB" title="Факториел — bulgara" lang="bg" hreflang="bg" data-title="Факториел" data-language-autonym="Български" data-language-local-name="bulgara" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A7%8C%E0%A6%A3%E0%A6%BF%E0%A6%95" title="গৌণিক — bengala" lang="bn" hreflang="bn" data-title="গৌণিক" data-language-autonym="বাংলা" data-language-local-name="bengala" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Faktorijel" title="Faktorijel — bosna" lang="bs" hreflang="bs" data-title="Faktorijel" data-language-autonym="Bosanski" data-language-local-name="bosna" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Factorial" title="Factorial — kataluna" lang="ca" hreflang="ca" data-title="Factorial" data-language-autonym="Català" data-language-local-name="kataluna" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%81%D8%A7%DA%A9%D8%AA%DB%86%D8%B1%DB%8C%DB%8E%D9%84" title="فاکتۆریێل — sorana" lang="ckb" hreflang="ckb" data-title="فاکتۆریێل" data-language-autonym="کوردی" data-language-local-name="sorana" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Faktori%C3%A1l" title="Faktoriál — ĉeĥa" lang="cs" hreflang="cs" data-title="Faktoriál" data-language-autonym="Čeština" data-language-local-name="ĉeĥa" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B0%D0%BB" title="Факториал — ĉuvaŝa" lang="cv" hreflang="cv" data-title="Факториал" data-language-autonym="Чӑвашла" data-language-local-name="ĉuvaŝa" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Fakultet_(matematik)" title="Fakultet (matematik) — dana" lang="da" hreflang="da" data-title="Fakultet (matematik)" data-language-autonym="Dansk" data-language-local-name="dana" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Fakult%C3%A4t_(Mathematik)" title="Fakultät (Mathematik) — germana" lang="de" hreflang="de" data-title="Fakultät (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="germana" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%B1%CF%81%CE%B1%CE%B3%CE%BF%CE%BD%CF%84%CE%B9%CE%BA%CF%8C" title="Παραγοντικό — greka" lang="el" hreflang="el" data-title="Παραγοντικό" data-language-autonym="Ελληνικά" data-language-local-name="greka" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="leginda artikolo"><a href="https://en.wikipedia.org/wiki/Factorial" title="Factorial — angla" lang="en" hreflang="en" data-title="Factorial" data-language-autonym="English" data-language-local-name="angla" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Factorial" title="Factorial — hispana" lang="es" hreflang="es" data-title="Factorial" data-language-autonym="Español" data-language-local-name="hispana" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Faktoriaal" title="Faktoriaal — estona" lang="et" hreflang="et" data-title="Faktoriaal" data-language-autonym="Eesti" data-language-local-name="estona" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Faktorial" title="Faktorial — eŭska" lang="eu" hreflang="eu" data-title="Faktorial" data-language-autonym="Euskara" data-language-local-name="eŭska" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%A7%DA%A9%D8%AA%D9%88%D8%B1%DB%8C%D9%84" title="فاکتوریل — persa" lang="fa" hreflang="fa" data-title="فاکتوریل" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kertoma" title="Kertoma — finna" lang="fi" hreflang="fi" data-title="Kertoma" data-language-autonym="Suomi" data-language-local-name="finna" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Factorielle" title="Factorielle — franca" lang="fr" hreflang="fr" data-title="Factorielle" data-language-autonym="Français" data-language-local-name="franca" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Factorial" title="Factorial — galega" lang="gl" hreflang="gl" data-title="Factorial" data-language-autonym="Galego" data-language-local-name="galega" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%A6%D7%A8%D7%AA_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="עצרת (מתמטיקה) — hebrea" lang="he" hreflang="he" data-title="עצרת (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="hebrea" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="क्रमगुणित — hinda" lang="hi" hreflang="hi" data-title="क्रमगुणित" data-language-autonym="हिन्दी" data-language-local-name="hinda" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Faktorijel" title="Faktorijel — kroata" lang="hr" hreflang="hr" data-title="Faktorijel" data-language-autonym="Hrvatski" data-language-local-name="kroata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Faktori%C3%A1lis" title="Faktoriális — hungara" lang="hu" hreflang="hu" data-title="Faktoriális" data-language-autonym="Magyar" data-language-local-name="hungara" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%96%D5%A1%D5%AF%D5%BF%D5%B8%D6%80%D5%AB%D5%A1%D5%AC" title="Ֆակտորիալ — armena" lang="hy" hreflang="hy" data-title="Ֆակտորիալ" data-language-autonym="Հայերեն" data-language-local-name="armena" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Factorial" title="Factorial — Interlingvao" lang="ia" hreflang="ia" data-title="Factorial" data-language-autonym="Interlingua" data-language-local-name="Interlingvao" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Faktorial" title="Faktorial — ibana" lang="iba" hreflang="iba" data-title="Faktorial" data-language-autonym="Jaku Iban" data-language-local-name="ibana" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Faktorial" title="Faktorial — indonezia" lang="id" hreflang="id" data-title="Faktorial" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezia" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Faktorialo" title="Faktorialo — Ido" lang="io" hreflang="io" data-title="Faktorialo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/A%C3%B0feldi" title="Aðfeldi — islanda" lang="is" hreflang="is" data-title="Aðfeldi" data-language-autonym="Íslenska" data-language-local-name="islanda" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Fattoriale" title="Fattoriale — itala" lang="it" hreflang="it" data-title="Fattoriale" data-language-autonym="Italiano" data-language-local-name="itala" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%9A%8E%E4%B9%97" title="階乗 — japana" lang="ja" hreflang="ja" data-title="階乗" data-language-autonym="日本語" data-language-local-name="japana" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%A4%E1%83%90%E1%83%A5%E1%83%A2%E1%83%9D%E1%83%A0%E1%83%98%E1%83%90%E1%83%9A%E1%83%98" title="მათემატიკური ფაქტორიალი — kartvela" lang="ka" hreflang="ka" data-title="მათემატიკური ფაქტორიალი" data-language-autonym="ქართული" data-language-local-name="kartvela" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B0%D0%BB" title="Факториал — kazaĥa" lang="kk" hreflang="kk" data-title="Факториал" data-language-autonym="Қазақша" data-language-local-name="kazaĥa" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%95%E0%B3%8D%E0%B2%B0%E0%B2%AE%E0%B2%97%E0%B3%81%E0%B2%A3%E0%B2%BF%E0%B2%A4" title="ಕ್ರಮಗುಣಿತ — kanara" lang="kn" hreflang="kn" data-title="ಕ್ರಮಗುಣಿತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kanara" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B3%84%EC%8A%B9_(%EC%88%98%ED%95%99)" title="계승 (수학) — korea" lang="ko" hreflang="ko" data-title="계승 (수학)" data-language-autonym="한국어" data-language-local-name="korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Factorialis" title="Factorialis — latino" lang="la" hreflang="la" data-title="Factorialis" data-language-autonym="Latina" data-language-local-name="latino" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Fatorial" title="Fatorial — Lombard" lang="lmo" hreflang="lmo" data-title="Fatorial" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Faktorialas" title="Faktorialas — litova" lang="lt" hreflang="lt" data-title="Faktorialas" data-language-autonym="Lietuvių" data-language-local-name="litova" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Faktori%C4%81ls" title="Faktoriāls — latva" lang="lv" hreflang="lv" data-title="Faktoriāls" data-language-autonym="Latviešu" data-language-local-name="latva" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B5%D0%BB" title="Факториел — makedona" lang="mk" hreflang="mk" data-title="Факториел" data-language-autonym="Македонски" data-language-local-name="makedona" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B5%8D%E0%B4%B0%E0%B4%AE%E0%B4%97%E0%B5%81%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82" title="ക്രമഗുണിതം — malajalama" lang="ml" hreflang="ml" data-title="ക്രമഗുണിതം" data-language-autonym="മലയാളം" data-language-local-name="malajalama" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%97%E0%A5%81%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="क्रमगुणित — marata" lang="mr" hreflang="mr" data-title="क्रमगुणित" data-language-autonym="मराठी" data-language-local-name="marata" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Faktorial" title="Faktorial — malaja" lang="ms" hreflang="ms" data-title="Faktorial" data-language-autonym="Bahasa Melayu" data-language-local-name="malaja" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Faculteit_(wiskunde)" title="Faculteit (wiskunde) — nederlanda" lang="nl" hreflang="nl" data-title="Faculteit (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="nederlanda" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fakultet_i_matematikk" title="Fakultet i matematikk — novnorvega" lang="nn" hreflang="nn" data-title="Fakultet i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="novnorvega" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Fakultet_(matematikk)" title="Fakultet (matematikk) — dannorvega" lang="nb" hreflang="nb" data-title="Fakultet (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="dannorvega" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Factoriala" title="Factoriala — okcitana" lang="oc" hreflang="oc" data-title="Factoriala" data-language-autonym="Occitan" data-language-local-name="okcitana" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%8D%E0%A8%B0%E0%A8%AE%E0%A8%97%E0%A9%81%E0%A8%A3%E0%A8%BF%E0%A8%A4" title="ਕ੍ਰਮਗੁਣਿਤ — panĝaba" lang="pa" hreflang="pa" data-title="ਕ੍ਰਮਗੁਣਿਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panĝaba" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Silnia" title="Silnia — pola" lang="pl" hreflang="pl" data-title="Silnia" data-language-autonym="Polski" data-language-local-name="pola" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Fatorial" title="Fatorial — Piedmontese" lang="pms" hreflang="pms" data-title="Fatorial" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fatorial" title="Fatorial — portugala" lang="pt" hreflang="pt" data-title="Fatorial" data-language-autonym="Português" data-language-local-name="portugala" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Factorial" title="Factorial — rumana" lang="ro" hreflang="ro" data-title="Factorial" data-language-autonym="Română" data-language-local-name="rumana" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B0%D0%BB" title="Факториал — rusa" lang="ru" hreflang="ru" data-title="Факториал" data-language-autonym="Русский" data-language-local-name="rusa" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Fatturiali" title="Fatturiali — sicilia" lang="scn" hreflang="scn" data-title="Fatturiali" data-language-autonym="Sicilianu" data-language-local-name="sicilia" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Faktorijel" title="Faktorijel — serbo-Kroata" lang="sh" hreflang="sh" data-title="Faktorijel" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-Kroata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9A%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B6%B8%E0%B7%8F%E0%B6%BB%E0%B7%9D%E0%B6%B4%E0%B7%92%E0%B6%AD%E0%B6%BA" title="ක්‍රමාරෝපිතය — sinhala" lang="si" hreflang="si" data-title="ක්‍රමාරෝපිතය" data-language-autonym="සිංහල" data-language-local-name="sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Factorial" title="Factorial — Simple English" lang="en-simple" hreflang="en-simple" data-title="Factorial" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Faktori%C3%A1l" title="Faktoriál — slovaka" lang="sk" hreflang="sk" data-title="Faktoriál" data-language-autonym="Slovenčina" data-language-local-name="slovaka" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Fakulteta_(funkcija)" title="Fakulteta (funkcija) — slovena" lang="sl" hreflang="sl" data-title="Fakulteta (funkcija)" data-language-autonym="Slovenščina" data-language-local-name="slovena" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Faktoriali" title="Faktoriali — albana" lang="sq" hreflang="sq" data-title="Faktoriali" data-language-autonym="Shqip" data-language-local-name="albana" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D1%98%D0%B5%D0%BB" title="Факторијел — serba" lang="sr" hreflang="sr" data-title="Факторијел" data-language-autonym="Српски / srpski" data-language-local-name="serba" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Fakultet_(matematik)" title="Fakultet (matematik) — sveda" lang="sv" hreflang="sv" data-title="Fakultet (matematik)" data-language-autonym="Svenska" data-language-local-name="sveda" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AF%8A%E0%AE%9F%E0%AE%B0%E0%AF%8D_%E0%AE%AA%E0%AF%86%E0%AE%B0%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%AE%E0%AF%8D" title="தொடர் பெருக்கம் — tamila" lang="ta" hreflang="ta" data-title="தொடர் பெருக்கம்" data-language-autonym="தமிழ்" data-language-local-name="tamila" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%9F%E0%B8%81%E0%B8%97%E0%B8%AD%E0%B9%80%E0%B8%A3%E0%B8%B5%E0%B8%A2%E0%B8%A5" title="แฟกทอเรียล — taja" lang="th" hreflang="th" data-title="แฟกทอเรียล" data-language-autonym="ไทย" data-language-local-name="taja" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Paktoryal" title="Paktoryal — tagaloga" lang="tl" hreflang="tl" data-title="Paktoryal" data-language-autonym="Tagalog" data-language-local-name="tagaloga" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Fakt%C3%B6riyel" title="Faktöriyel — turka" lang="tr" hreflang="tr" data-title="Faktöriyel" data-language-autonym="Türkçe" data-language-local-name="turka" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D1%96%D0%B0%D0%BB" title="Факторіал — ukraina" lang="uk" hreflang="uk" data-title="Факторіал" data-language-autonym="Українська" data-language-local-name="ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B9%D8%A7%D9%85%D9%84%DB%8C%DB%81" title="عاملیہ — urduo" lang="ur" hreflang="ur" data-title="عاملیہ" data-language-autonym="اردو" data-language-local-name="urduo" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Faktorial" title="Faktorial — uzbeka" lang="uz" hreflang="uz" data-title="Faktorial" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeka" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Giai_th%E1%BB%ABa" title="Giai thừa — vjetnama" lang="vi" hreflang="vi" data-title="Giai thừa" data-language-autonym="Tiếng Việt" data-language-local-name="vjetnama" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%98%B6%E4%B9%98" title="阶乘 — vua" lang="wuu" hreflang="wuu" data-title="阶乘" data-language-autonym="吴语" data-language-local-name="vua" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%9A%8E%E4%B9%98" title="階乘 — ĉina" lang="zh" hreflang="zh" data-title="階乘" data-language-autonym="中文" data-language-local-name="ĉina" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E9%9A%8E%E4%B9%98" title="階乘 — Literary Chinese" lang="lzh" hreflang="lzh" data-title="階乘" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%9A%8E%E4%B9%98" title="階乘 — kantona" lang="yue" hreflang="yue" data-title="階乘" data-language-autonym="粵語" data-language-local-name="kantona" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q120976#sitelinks-wikipedia" title="Redakti interlingvajn ligilojn" class="wbc-editpage">Redakti ligilojn</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Nomspacoj"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li 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vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Iloj</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">movigi al flanka panelo</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">kaŝi</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Pliaj opcioj" > <div class="vector-menu-heading"> Agoj </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Faktorialo"><span>Legi</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Faktorialo&veaction=edit" title="Redakti ĉi tiun paĝon [v]" accesskey="v"><span>Redakti</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Faktorialo&action=edit" title="Redakti fontan kodon de tiu ĉi paĝo [e]" accesskey="e"><span>Redakti fonton</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Faktorialo&action=history"><span>Vidi historion</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Ĝenerale </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speciala%C4%B5o:Kio_ligas_%C4%89i_tien%3F/Faktorialo" title="Listo de ĉiuj vikiaj paĝoj kiuj ligas ĉi tien [j]" accesskey="j"><span>Ligiloj ĉi tien</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speciala%C4%B5o:Rilataj_%C5%9Dan%C4%9Doj/Faktorialo" rel="nofollow" title="Lastaj ŝanĝoj en paĝoj kiuj ligas al ĉi tiu paĝo [k]" accesskey="k"><span>Rilataj ŝanĝoj</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=eo" title="Alŝuti dosierojn [u]" accesskey="u"><span>Alŝuti dosieron</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speciala%C4%B5o:Specialaj_pa%C4%9Doj" title="Listo de ĉiuj specialaj paĝoj [q]" accesskey="q"><span>Specialaj paĝoj</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Faktorialo&oldid=8762704" title="Porĉiama ligilo al ĉi versio de la paĝo"><span>Konstanta ligilo</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Faktorialo&action=info" title="Pli da informo pri ĉi tiu paĝo"><span>Informoj pri la paĝo</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speciala%C4%B5o:Citi&page=Faktorialo&id=8762704&wpFormIdentifier=titleform" title="Informoj pri kiel citi ĉi tiun paĝon"><span>Citi ĉi tiun artikolon</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speciala%C4%B5o:UrlShortener&url=https%3A%2F%2Feo.wikipedia.org%2Fwiki%2FFaktorialo"><span>Akiri mallongigitan URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Speciala%C4%B5o:QrCode&url=https%3A%2F%2Feo.wikipedia.org%2Fwiki%2FFaktorialo"><span>Elŝutu QR-kodon</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Presi/elporti </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Speciala%C4%B5o:Libro&bookcmd=book_creator&referer=Faktorialo"><span>Krei libron</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speciala%C4%B5o:DownloadAsPdf&page=Faktorialo&action=show-download-screen"><span>Elŝuti kiel PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Faktorialo&printable=yes" title="Presebla versio de tiu ĉi paĝo [p]" accesskey="p"><span>Presebla versio</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> En aliaj projektoj </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Factorial_(function)" hreflang="en"><span>Vikimedia Komunejo</span></a></li><li class="wb-otherproject-link wb-otherproject-wikifunctions mw-list-item"><a href="https://www.wikifunctions.org/wiki/Z13667" hreflang="en"><span>Wikifunctions</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q120976" title="Ligilo al konektita datuma ero [g]" accesskey="g"><span>Ero en Vikidatumoj</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Paĝaj iloj"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspekto"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aspekto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">movigi al flanka panelo</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">kaŝi</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">El Vikipedio, la libera enciklopedio</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="eo" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/Dosiero:Log-factorial.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Log-factorial.svg/250px-Log-factorial.svg.png" decoding="async" width="250" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Log-factorial.svg/375px-Log-factorial.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Log-factorial.svg/500px-Log-factorial.svg.png 2x" data-file-width="600" data-file-height="480" /></a><figcaption>Grafikaĵo de <a href="/wiki/Logaritmo" title="Logaritmo">logaritmo</a> de faktorialo kontraŭ logaritmo de la argumento</figcaption></figure> <div class="thumb tright"> <table class="wikitable"> <tbody><tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> </th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"></span> </th></tr> <tr> <td>0</td> <td>1 </td></tr> <tr> <td>1</td> <td>1 </td></tr> <tr> <td>2</td> <td>2 </td></tr> <tr> <td>3</td> <td>6 </td></tr> <tr> <td>4</td> <td>24 </td></tr> <tr> <td>5</td> <td>120 </td></tr> <tr> <td>6</td> <td>720 </td></tr> <tr> <td>7</td> <td>5040 </td></tr> <tr> <td>8</td> <td>40320 </td></tr> <tr> <td>9</td> <td>362880 </td></tr> <tr> <td>10</td> <td>3 628 800 </td></tr> <tr> <td>11</td> <td>39 916 800 </td></tr> <tr> <td>12</td> <td>479 001 600 </td></tr> <tr> <td>13</td> <td>6 227 020 800 </td></tr> <tr> <td>14</td> <td>87 178 291 200 </td></tr> <tr> <td>15</td> <td>1 307 674 368 000 </td></tr> <tr> <td>20</td> <td>2 432 902 008 176 640 000 </td></tr> <tr> <td>25</td> <td>15 511 210 043 330 985 984 000 000 </td></tr> <tr> <td>50</td> <td>3,04140932... × 10<sup>64</sup> </td></tr> <tr> <td>70</td> <td>1,19785717... × <a href="/wiki/Guglo" title="Guglo">10<sup>100</sup></a> </td></tr> <tr> <td>450</td> <td>1,73336873... × 10<sup>1000</sup> </td></tr> <tr> <td>3249</td> <td>6,41233768... × 10<sup>10000</sup> </td></tr> <tr> <td>25206</td> <td>1,205703438... × 10<sup>100000</sup> </td></tr> <tr> <td>47176</td> <td>8,4485731495... × 10<sup>200001</sup> </td></tr> <tr> <td>100000</td> <td>2,8242294079... × 10<sup>456573</sup> </td></tr></tbody></table> </div> <p>En la <a href="/wiki/Matematiko" title="Matematiko">matematiko</a>, <b>faktorialo</b> de <a href="/wiki/Natura_nombro" title="Natura nombro">natura nombro</a> <i>n</i> estas la <a href="/wiki/Produto" title="Produto">produto</a> de la <a href="/wiki/Pozitiva" class="mw-disambig" title="Pozitiva">pozitivaj</a> <a href="/wiki/Entjero" title="Entjero">entjeroj</a> malpli aŭ egalaj al <i>n</i>. Oni signas ĝin per <i>n!</i>, kion oni prononcas <i>no faktoriale</i> laŭ <a href="/wiki/Christian_Kramp" title="Christian Kramp">Christian Kramp</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Difino">Difino</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorialo&veaction=edit&section=1" title="Redakti sekcion: Difino" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorialo&action=edit&section=1" title="Redakti la fontkodo de la sekcio: Difino"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!=1\cdot 2\cdot 3\cdots (n-1)\cdot n=\prod _{k=1}^{n}k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!=1\cdot 2\cdot 3\cdots (n-1)\cdot n=\prod _{k=1}^{n}k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546f4ddf968be1454ade2f613d39b148054cdccd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.43ex; height:6.843ex;" alt="{\displaystyle n!=1\cdot 2\cdot 3\cdots (n-1)\cdot n=\prod _{k=1}^{n}k}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!=n\cdot (n-1)!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>=</mo> <mi>n</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!=n\cdot (n-1)!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09d762bf72a3e9d56651aabaa571f98cf777e19a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.068ex; height:2.843ex;" alt="{\displaystyle n!=n\cdot (n-1)!}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0!=1!=1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>!</mo> <mo>=</mo> <mn>1</mn> <mo>!</mo> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0!=1!=1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb09b9bda87e70414633c7f069d0dc18f466b7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.365ex; height:2.176ex;" alt="{\displaystyle 0!=1!=1\,}"></span></dd></dl> <p>Oni aldone difinas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0!=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>!</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0!=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22956a0fa255c6c9562eab440f8c23c2954a6cf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.07ex; height:2.176ex;" alt="{\displaystyle 0!=1}"></span>, ĉar ĝenerale la <a href="/wiki/Produto" title="Produto">produto</a> de neniuj <a href="/wiki/Faktoro" class="mw-redirect" title="Faktoro">faktoroj</a> estas konsiderata 1. </p> <div class="mw-heading mw-heading2"><h2 id="Kombinatoriko">Kombinatoriko</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorialo&veaction=edit&section=2" title="Redakti sekcion: Kombinatoriko" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorialo&action=edit&section=2" title="Redakti la fontkodo de la sekcio: Kombinatoriko"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En <a href="/wiki/Kombinatoriko" title="Kombinatoriko">kombinatoriko</a>, faktorialo <i>n!</i> estas kvanto de <a href="/wiki/Permuta%C4%B5o" title="Permutaĵo">permutaĵoj</a> de <i>n</i> eroj. Ekzemple: </p> <dl><dd>Por 1 ero {<i>A</i>} estas 1!=1 permuto: <dl><dd>A</dd></dl></dd> <dd>Por 2 eroj {<i>A</i>,<i>B</i>} estas 2!=2 permutaĵoj: <dl><dd>AB   BA</dd></dl></dd> <dd>Por 3 eroj {<i>A</i>,<i>B</i>,<i>C</i>} estas 3!=6 permutaĵoj: <dl><dd>ABC   ACB   BAC   BCA   CAB   CBA</dd></dl></dd> <dd>Por 4 eroj {<i>A</i>,<i>B</i>,<i>C</i>,<i>D</i>} estas 4!=24 permutaĵoj: <dl><dd>ABCD   BACD   CABD   DABC</dd> <dd>ABDC   BADC   CADB   DACB</dd> <dd>ACBD   BCAD   CBAD   DBAC</dd> <dd>ACDB   BCDA   CBDA   DBCA</dd> <dd>ADBC   BDAC   CDAB   DCAB</dd> <dd>ADCB   BDCA   CDBA   DCBA</dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Γ-funkcio"><span id=".CE.93-funkcio"></span>Γ-funkcio</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorialo&veaction=edit&section=3" title="Redakti sekcion: Γ-funkcio" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorialo&action=edit&section=3" title="Redakti la fontkodo de la sekcio: Γ-funkcio"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%CE%93-funkcio" title="Γ-funkcio">Γ-funkcio</a> estas <a href="/wiki/Funkcio_(matematiko)" title="Funkcio (matematiko)">funkcio</a>, difinita por ĉiuj <a href="/wiki/Reela_nombro" class="mw-redirect" title="Reela nombro">reelaj</a> aŭ <a href="/wiki/Kompleksa_nombro" title="Kompleksa nombro">kompleksaj</a> argumentoj krom nepozitivaj entjeroj (0, -1, -2, -3, ...). Ĝi estas vastigaĵo de faktorialo. Se <i>n</i> estas <a href="/wiki/Nenegativa" class="mw-redirect" title="Nenegativa">nenegativa</a> <a href="/wiki/Entjero" title="Entjero">entjero</a> (0, 1, 2, 3, ...), do </p> <dl><dd><i>Γ(n+1) = n!</i></dd></dl> <p>Aŭ ekvivalente se <i>n</i> estas <a href="/wiki/Pozitiva" class="mw-disambig" title="Pozitiva">pozitiva</a> <a href="/wiki/Entjero" title="Entjero">entjero</a> (1, 2, 3, 4, ...), do </p> <dl><dd><i>Γ(n) = (n-1)!</i></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Proksimuma_kalkulado_de_Stirling">Proksimuma kalkulado de Stirling</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorialo&veaction=edit&section=4" title="Redakti sekcion: Proksimuma kalkulado de Stirling" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorialo&action=edit&section=4" title="Redakti la fontkodo de la sekcio: Proksimuma kalkulado de Stirling"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/w/index.php?title=Proksimuma_kalkulado_de_Stirling&action=edit&redlink=1" class="new" title="Proksimuma kalkulado de Stirling (paĝo ne ekzistas)">Proksimuma kalkulado de Stirling</a> estas proksimuma formulo por faktoriala: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+{\frac {1}{12n}}+{\frac {1}{288n^{2}}}-{\frac {139}{51840n^{3}}}+O\left(n^{-4}\right)\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <mi>n</mi> </msqrt> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>e</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>12</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>288</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>139</mn> <mrow> <mn>51840</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>O</mi> <mrow> <mo>(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+{\frac {1}{12n}}+{\frac {1}{288n^{2}}}-{\frac {139}{51840n^{3}}}+O\left(n^{-4}\right)\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a46e72d25eaa558dbbaf6fa9af5bbcdc01312a5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.999ex; height:6.176ex;" alt="{\displaystyle n!={\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\left(1+{\frac {1}{12n}}+{\frac {1}{288n^{2}}}-{\frac {139}{51840n^{3}}}+O\left(n^{-4}\right)\right),}"></span></dd></dl> <p>kie la nombro <i><a href="/wiki/E_(matematiko)" title="E (matematiko)">e</a></i> estas la bazo de la <a href="/wiki/Eksponenta_funkcio" title="Eksponenta funkcio">eksponenta funkcio</a> kaj <i>O</i> estas <a href="/wiki/Granda_O" title="Granda O">granda O</a>. </p><p>Pli simpla, malpli preciza sed iam uzebla estas formulo kun nur la unua membro de la proksimuma kalkulado de Stirling </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!\approx {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <mi>n</mi> </msqrt> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>e</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!\approx {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7136fede6a5ec27a99e73f47a17234df0f4bd94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.19ex; height:4.843ex;" alt="{\displaystyle n!\approx {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}"></span></dd></dl> <p>Tiam estas limigoj por la faktorialo. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}<n!<{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}e^{1/(12n)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <mi>n</mi> </msqrt> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>e</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo><</mo> <mi>n</mi> <mo>!</mo> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <mi>n</mi> </msqrt> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>e</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>12</mn> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}<n!<{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}e^{1/(12n)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dccfce03095a19639d4cb79a7335903410893061" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.207ex; height:4.843ex;" alt="{\displaystyle {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}<n!<{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}e^{1/(12n)}}"></span></dd></dl> <p>Tia <a href="/wiki/Proksimuma_kalkulado" title="Proksimuma kalkulado">proksimumo</a> permesas ankaŭ trovi proksimumon pri la <a href="/wiki/Logaritmo" title="Logaritmo">logaritmo</a> de n! : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \left(n!\right)\simeq n\ln \left(n\right)-n\ \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mo>)</mo> </mrow> <mo>≃</mo> <mi>n</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>−</mo> <mi>n</mi> <mtext> </mtext> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \left(n!\right)\simeq n\ln \left(n\right)-n\ \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55989cff5d2585a3a1cc97d94c57470226bd7513" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.664ex; height:2.843ex;" alt="{\displaystyle \ln \left(n!\right)\simeq n\ln \left(n\right)-n\ \,.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Duopa_faktorialo">Duopa faktorialo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorialo&veaction=edit&section=5" title="Redakti sekcion: Duopa faktorialo" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorialo&action=edit&section=5" title="Redakti la fontkodo de la sekcio: Duopa faktorialo"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Duopa faktorialo</b> estas: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!!={\begin{cases}1,&{\mbox{ se }}n=0{\mbox{ aŭ }}n=1;\\n\times (n-2)!!&{\mbox{ se }}n\geq 2.\qquad \qquad \end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>!</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> se </mtext> </mstyle> </mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> aŭ </mtext> </mstyle> </mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mi>n</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> se </mtext> </mstyle> </mrow> <mi>n</mi> <mo>≥</mo> <mn>2.</mn> <mspace width="2em" /> <mspace width="2em" /> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!!={\begin{cases}1,&{\mbox{ se }}n=0{\mbox{ aŭ }}n=1;\\n\times (n-2)!!&{\mbox{ se }}n\geq 2.\qquad \qquad \end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba54799e6b265a46ecdfa9fee74d008432fe1742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:42.148ex; height:6.509ex;" alt="{\displaystyle n!!={\begin{cases}1,&{\mbox{ se }}n=0{\mbox{ aŭ }}n=1;\\n\times (n-2)!!&{\mbox{ se }}n\geq 2.\qquad \qquad \end{cases}}}"></span></dd></dl> <p>Tiel: </p> <dl><dd><i>n!!=n(n-2)(n-4)·...·6·4·2</i> se <i>n</i> estas <a href="/wiki/Pareco_de_nombroj" title="Pareco de nombroj">para</a> pozitiva;</dd> <dd><i>n!!=n(n-2)(n-4)·...·5·3·1</i> se <i>n</i> estas <a href="/wiki/Pareco_de_nombroj" title="Pareco de nombroj">nepara</a> pozitiva;</dd></dl> <p>Notu, ke duopa faktorialo ne estas faktorialo de faktorialo, ĝenerale <i>n!!≠(n!)!</i>. </p><p>La difino povas esti etendita reen al la negativaj argumentoj ĉar </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-2)!!={\frac {n!!}{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> <mo>!</mo> <mo>!</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> <mo>!</mo> </mrow> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-2)!!={\frac {n!!}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb9b1eea666ff0b0c5144e81bdd5965cd53c73e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.124ex; height:5.343ex;" alt="{\displaystyle (n-2)!!={\frac {n!!}{n}}}"></span></dd></dl> <p>Tiel: </p> <dl><dd><i>n!!=1/( (n+2)(n+4)·...·(-3)·(-1)·1 )</i> se <i>n</i> estas nepara negativa.</dd></dl> <p>Per ĉi tia maniero duopa faktorialo ne estas difinita por para negativa argumento, tamen vidu sube pri ebleco difini per <a href="/wiki/%CE%93_funkcio" class="mw-redirect" title="Γ funkcio">Γ funkcio</a>. </p><p>Ekzemple, 8!! = 2 · 4 · 6 · 8 = 384, 9!! = 1 · 3 · 5 · 7 · 9 = 945. </p><p>Valoroj de <i>n!!</i> por <i>n=0, 1, 2, ...</i> estas: </p> <dl><dd>1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, ...</dd></dl> <p>Valoroj de <i>n!!</i> por <i>n=-1, -3, -5, ...</i> estas: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,-1,{\tfrac {1}{3}},-{\tfrac {1}{15}},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>15</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,-1,{\tfrac {1}{3}},-{\tfrac {1}{15}},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69658487e63de9796531c0a51cd8c23c2a7e2e20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.938ex; height:3.676ex;" alt="{\displaystyle 1,-1,{\tfrac {1}{3}},-{\tfrac {1}{15}},\dots }"></span></dd></dl> <p>Iuj formuloj kun duopa faktorialo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!=n!!(n-1)!!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>=</mo> <mi>n</mi> <mo>!</mo> <mo>!</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!=n!!(n-1)!!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482312f10337ad46d6cdca97cfeb1ea876f271a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.329ex; height:2.843ex;" alt="{\displaystyle n!=n!!(n-1)!!}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2n)!!=2^{n}n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2n)!!=2^{n}n!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85bee36bef8000f5ead8dbb713edeb37eceab1fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.181ex; height:2.843ex;" alt="{\displaystyle (2n)!!=2^{n}n!}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2n+1)!!={(2n+1)! \over (2n)!!}={(2n+1)! \over 2^{n}n!}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2n+1)!!={(2n+1)! \over (2n)!!}={(2n+1)! \over 2^{n}n!}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd818f59432abf2064c0b394c4352b241c4e1535" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.565ex; height:6.509ex;" alt="{\displaystyle (2n+1)!!={(2n+1)! \over (2n)!!}={(2n+1)! \over 2^{n}n!}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2n-1)!!={(2n-1)! \over (2n-2)!!}={(2n)! \over 2^{n}n!}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2n-1)!!={(2n-1)! \over (2n-2)!!}={(2n)! \over 2^{n}n!}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/752a4c7acc1bf76e1d6d289fb1b3af7e2f2926f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.209ex; height:6.509ex;" alt="{\displaystyle (2n-1)!!={(2n-1)! \over (2n-2)!!}={(2n)! \over 2^{n}n!}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \left(n+{1 \over 2}\right)={\sqrt {\pi }}\,\,{(2n-1)!! \over 2^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \left(n+{1 \over 2}\right)={\sqrt {\pi }}\,\,{(2n-1)!! \over 2^{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/149d1da605834aa2e48e70db1a7d1e7c06e2166d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.134ex; height:6.343ex;" alt="{\displaystyle \Gamma \left(n+{1 \over 2}\right)={\sqrt {\pi }}\,\,{(2n-1)!! \over 2^{n}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma \left({n \over 2}+1\right)={\sqrt {\pi }}\,\,{n!! \over 2^{(n+1)/2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> <mo>!</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma \left({n \over 2}+1\right)={\sqrt {\pi }}\,\,{n!! \over 2^{(n+1)/2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae239ef795169f674f796c1918f2749ac4d6f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.231ex; height:6.009ex;" alt="{\displaystyle \Gamma \left({n \over 2}+1\right)={\sqrt {\pi }}\,\,{n!! \over 2^{(n+1)/2}}}"></span></dd></dl> <p>kie <i>Γ</i> estas <a href="/wiki/%CE%93_funkcio" class="mw-redirect" title="Γ funkcio">Γ funkcio</a>. La lasta formulo povas esti konsiderata kiel difino de duopa faktorialo por ĉiuj kompleksaj <i>n≠0</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Plurfaktorialo">Plurfaktorialo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorialo&veaction=edit&section=6" title="Redakti sekcion: Plurfaktorialo" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorialo&action=edit&section=6" title="Redakti la fontkodo de la sekcio: Plurfaktorialo"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Plurfaktorialo</b> estas plua ĝeneraligo post la duopa faktorialo. Plurfaktorialo de la <i>k</i>-a ordo de <i>n</i>, aŭ alivorte la <i>k</i>-a plurfaktorialo de <i>n</i>, estas </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!^{(k)}=\left\{{\begin{matrix}1,\qquad \qquad \ &&{\mbox{se }}0\leq n<k;\\n(n-k)!^{(k)},&&{\mbox{se }}n\geq k.\quad \ \ \,\end{matrix}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <msup> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mo>,</mo> <mspace width="2em" /> <mspace width="2em" /> <mtext> </mtext> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>se </mtext> </mstyle> </mrow> <mn>0</mn> <mo>≤</mo> <mi>n</mi> <mo><</mo> <mi>k</mi> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>se </mtext> </mstyle> </mrow> <mi>n</mi> <mo>≥</mo> <mi>k</mi> <mo>.</mo> <mspace width="1em" /> <mtext> </mtext> <mtext> </mtext> <mspace width="thinmathspace" /> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!^{(k)}=\left\{{\begin{matrix}1,\qquad \qquad \ &&{\mbox{se }}0\leq n<k;\\n(n-k)!^{(k)},&&{\mbox{se }}n\geq k.\quad \ \ \,\end{matrix}}\right.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdbda36a16bba50c233116c04d6aa0f1400b4a67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.101ex; height:6.509ex;" alt="{\displaystyle n!^{(k)}=\left\{{\begin{matrix}1,\qquad \qquad \ &&{\mbox{se }}0\leq n<k;\\n(n-k)!^{(k)},&&{\mbox{se }}n\geq k.\quad \ \ \,\end{matrix}}\right.}"></span></dd></dl> <p>Duopa faktorialo estas plurfaktorialo de la <i>2</i>-a ordo. </p> <div class="mw-heading mw-heading2"><h2 id="Primofaktorialo">Primofaktorialo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorialo&veaction=edit&section=7" title="Redakti sekcion: Primofaktorialo" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorialo&action=edit&section=7" title="Redakti la fontkodo de la sekcio: Primofaktorialo"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Primofaktorialo" title="Primofaktorialo">Primofaktorialo</a></b> <i>n#</i> estas produto de ĉiuj <a href="/wiki/Primo" title="Primo">primoj</a> ne pli grandaj ol <i>n</i>. Ekzemple: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 11\#=12\#=2\cdot 3\cdot 5\cdot 7\cdot 11=2310}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>11</mn> <mi mathvariant="normal">#</mi> <mo>=</mo> <mn>12</mn> <mi mathvariant="normal">#</mi> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> <mo>⋅</mo> <mn>7</mn> <mo>⋅</mo> <mn>11</mn> <mo>=</mo> <mn>2310</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 11\#=12\#=2\cdot 3\cdot 5\cdot 7\cdot 11=2310}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9da76981ef317eaf588a0b488dcbd071a02a24f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.158ex; height:2.509ex;" alt="{\displaystyle 11\#=12\#=2\cdot 3\cdot 5\cdot 7\cdot 11=2310}"></span></dd></dl> <p>Se <i>p<sub>n</sub></i> estas la <i>n</i>-a primo, do <i>p<sub>n</sub>#</i> estas produto de <i>n</i> la unuaj primoj: </p><p>La unuaj valoroj de <i>p<sub>n</sub>#</i> por <i>n=1, 2, 3, ...</i> estas: </p> <dl><dd><a href="/wiki/Du" class="mw-redirect" title="Du">2</a>, <a href="/wiki/Ses" class="mw-redirect" title="Ses">6</a>, <a href="/wiki/Tridek" class="mw-redirect" title="Tridek">30</a>, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Vidu_ankaŭ"><span id="Vidu_anka.C5.AD"></span>Vidu ankaŭ</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorialo&veaction=edit&section=8" title="Redakti sekcion: Vidu ankaŭ" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorialo&action=edit&section=8" title="Redakti la fontkodo de la sekcio: Vidu ankaŭ"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Kombinatoriko" title="Kombinatoriko">Kombinatoriko</a></li> <li><a href="/wiki/%CE%93-funkcio" title="Γ-funkcio">Γ-funkcio</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Eksteraj_ligiloj">Eksteraj ligiloj</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorialo&veaction=edit&section=9" title="Redakti sekcion: Eksteraj ligiloj" class="mw-editsection-visualeditor"><span>redakti</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorialo&action=edit&section=9" title="Redakti la fontkodo de la sekcio: Eksteraj ligiloj"><span>redakti fonton</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint" id="aliaj_projektoj" style="float: right; clear: right; width: 250px; border: 1px solid #aaa; padding: 4px; font-size: 90%; background: #f9f9f9;"> <ul class="noarchive"><li class="commons"><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Factorial_(function)?uselang=eo">Kategorio Faktorialo en la <span>Vikimedia Komunejo</span></a> (Multrimedaj datumoj)</span> </li></ul> </div> <ul><li><a rel="nofollow" class="external text" href="http://factorielle.free.fr">Retejo pri faktorialo</a> (cs, en, fr)</li> <li><a rel="nofollow" class="external text" href="//oeis.org/A000142">A000142</a> en <a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a> - valoroj de faktorialo <i>n!</i> por <i>n = 0, 1, 2, ...</i></li> <li><a rel="nofollow" class="external text" href="//oeis.org/A006882">A006882</a> en <a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a> - valoroj de duopa faktorialo <i>n!!</i> por <i>n = 0, 1, 2, ...</i></li> <li><a rel="nofollow" class="external text" href="//oeis.org/A002110">A002110</a> en <a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a> - valoroj de <i>p<sub>n</sub>#</i></li> <li><a rel="nofollow" class="external text" href="//oeis.org/A001163">A001163</a> en <a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a> - <a href="/wiki/Numeratoro" class="mw-redirect" title="Numeratoro">numeratoroj</a> de <a href="/w/index.php?title=Proksimuma_kalkulado_de_Stirling&action=edit&redlink=1" class="new" title="Proksimuma kalkulado de Stirling (paĝo ne ekzistas)">proksimuma kalkulado de Stirling</a></li> <li><a rel="nofollow" class="external text" href="//oeis.org/A001164">A001164</a> en <a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a> - <a href="/wiki/Denominatoro" class="mw-redirect" title="Denominatoro">denominatoroj</a> de proksimuma kalkulado de Stirling</li></ul></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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