Stirling numbers of the first kind

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interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_Stirlinga#Liczby_Stirlinga_I_rodzaju" title="Liczby Stirlinga – Polish" lang="pl" hreflang="pl" data-title="Liczby Stirlinga" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%A1%D1%82%D0%B8%D1%80%D0%BB%D0%B8%D0%BD%D0%B3%D0%B0_%D0%BF%D0%B5%D1%80%D0%B2%D0%BE%D0%B3%D0%BE_%D1%80%D0%BE%D0%B4%D0%B0" title="Числа Стирлинга первого рода – Russian" lang="ru" hreflang="ru" data-title="Числа Стирлинга первого рода" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a 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free encyclopedia</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="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Count of permutations by cycles</div> <p> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, especially in <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>, <b>Stirling numbers of the first kind</b> arise in the study of permutations. In particular, the unsigned Stirling numbers of the first kind count <a href="/wiki/Permutation" title="Permutation">permutations</a> according to their number of <a href="/wiki/Cycles_and_fixed_points" title="Cycles and fixed points">cycles</a> (counting <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed points</a> as cycles of length one).<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The Stirling numbers of the first and <a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">second kind</a> can be understood as inverses of one another when viewed as <a href="/wiki/Triangular_matrix" title="Triangular matrix">triangular matrices</a>. This article is devoted to specifics of Stirling numbers of the first kind. Identities linking the two kinds appear in the article on <a href="/wiki/Stirling_number" title="Stirling number">Stirling numbers</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Definition_by_algebra">Definition by algebra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=2" title="Edit section: Definition by algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Signed Stirling numbers of the first kind are the coefficients <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 s(n,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(n,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a462712cd2a4765947d2ca485db7b9d86cdc370f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.54ex; height:2.843ex;" alt="{\displaystyle s(n,k)}"></span> in the expansion of the <a href="/wiki/Falling_factorial" class="mw-redirect" title="Falling factorial">falling factorial</a> </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 (x)_{n}=x(x-1)(x-2)\cdots (x-n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x)_{n}=x(x-1)(x-2)\cdots (x-n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6342c004cefacfef11716c6a2608f093dfc2018" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.944ex; height:2.843ex;" alt="{\displaystyle (x)_{n}=x(x-1)(x-2)\cdots (x-n+1)}"></span></dd></dl> <p>into powers of the variable <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>: </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 (x)_{n}=\sum _{k=0}^{n}s(n,k)x^{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <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> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x)_{n}=\sum _{k=0}^{n}s(n,k)x^{k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/052f2b17a59ee5e7ac8d74404fa9414f8f2deaa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.803ex; height:7.009ex;" alt="{\displaystyle (x)_{n}=\sum _{k=0}^{n}s(n,k)x^{k},}"></span></dd></dl> <p>For example, <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 (x)_{3}=x(x-1)(x-2)=x^{3}-3x^{2}+2x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x)_{3}=x(x-1)(x-2)=x^{3}-3x^{2}+2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1898e9a9e8f1097d098a3c1f16c8628bc03e286" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.107ex; height:3.176ex;" alt="{\displaystyle (x)_{3}=x(x-1)(x-2)=x^{3}-3x^{2}+2x}"></span>, leading to the values <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 s(3,3)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(3,3)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d86adbcf2f74916f2c4a265243da92833016b56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.52ex; height:2.843ex;" alt="{\displaystyle s(3,3)=1}"></span>, <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 s(3,2)=-3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(3,2)=-3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2128486611e5cb6a7bc18a6578ae015558989383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.328ex; height:2.843ex;" alt="{\displaystyle s(3,2)=-3}"></span>, and <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 s(3,1)=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(3,1)=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e9b72502efde2203a54834a15650a00f24e56fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.52ex; height:2.843ex;" alt="{\displaystyle s(3,1)=2}"></span>. </p><p>The unsigned Stirling numbers may also be defined algebraically as the coefficients of the <a href="/wiki/Rising_factorial" class="mw-redirect" title="Rising factorial">rising factorial</a>: </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 x^{\overline {n}}=x(x+1)\cdots (x+n-1)=\sum _{k=0}^{n}\left[{n \atop k}\right]x^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo accent="false">¯</mo> </mover> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <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>n</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\overline {n}}=x(x+1)\cdots (x+n-1)=\sum _{k=0}^{n}\left[{n \atop k}\right]x^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c402717f587efa42aa2353e6679ecbf49756c432" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.879ex; height:7.009ex;" alt="{\displaystyle x^{\overline {n}}=x(x+1)\cdots (x+n-1)=\sum _{k=0}^{n}\left[{n \atop k}\right]x^{k}}"></span>.</dd></dl> <p>The notations used on this page for Stirling numbers are not universal, and may conflict with notations in other sources; the square bracket notation <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 \left[{n \atop k}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{n \atop k}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e244be41a40ffd185e9edea632ba557f6ad945" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.147ex; height:4.843ex;" alt="{\displaystyle \left[{n \atop k}\right]}"></span> is also common notation for the <a href="/wiki/Gaussian_binomial_coefficient" title="Gaussian binomial coefficient">Gaussian coefficients</a>.<sup id="cite_ref-:1_2-0" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Definition_by_permutation">Definition by permutation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=3" title="Edit section: Definition by permutation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Subsequently, it was discovered that the absolute values <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 |s(n,k)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |s(n,k)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c3379235f42ad60f53b76dcdb3841f898b7274c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.833ex; height:2.843ex;" alt="{\displaystyle |s(n,k)|}"></span> of these numbers are equal to the number of <a href="/wiki/Permutation" title="Permutation">permutations</a> of certain kinds. These absolute values, which are known as unsigned Stirling numbers of the first kind, are often denoted <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 c(n,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(n,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b386a195c7e3c04b1d859ebbe0caf61cedbc144c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.456ex; height:2.843ex;" alt="{\displaystyle c(n,k)}"></span> or <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 \left[{n \atop k}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{n \atop k}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e244be41a40ffd185e9edea632ba557f6ad945" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.147ex; height:4.843ex;" alt="{\displaystyle \left[{n \atop k}\right]}"></span>. They may be defined directly to be the number of <a href="/wiki/Permutation" title="Permutation">permutations</a> of <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> elements with <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> disjoint <a href="/wiki/Cyclic_permutation" title="Cyclic permutation">cycles</a>.<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p> For example, of the <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 3!=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>!</mo> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3!=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67961712fc0a2db61a7afc54c8d0ca99c73ff3bd" 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 3!=6}"></span> permutations of three elements, there is one permutation with three cycles (the <a href="/wiki/Identity_permutation" class="mw-redirect" title="Identity permutation">identity permutation</a>, given in <a href="/wiki/Permutation#Cycle_notation" title="Permutation">one-line notation</a> by <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 123}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>123</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 123}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfb9d5d9dcca702fd44d4463293cb15396429a68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 123}"></span> or in <a href="/wiki/Cycle_notation" class="mw-redirect" title="Cycle notation">cycle notation</a> by <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)(2)(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1)(2)(3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d79151e044b979ff33af3cc79a05095c4237269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.915ex; height:2.843ex;" alt="{\displaystyle (1)(2)(3)}"></span>), three permutations with two cycles (<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 132=(1)(23)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>132</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>23</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 132=(1)(23)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/996498b378520a417b72c018980b40810df96519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.692ex; height:2.843ex;" alt="{\displaystyle 132=(1)(23)}"></span>, <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 213=(12)(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>213</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>12</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 213=(12)(3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee3a92e4685d659eb034681c7a3d97f9717c11e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.692ex; height:2.843ex;" alt="{\displaystyle 213=(12)(3)}"></span>, and <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 321=(13)(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>321</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>13</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 321=(13)(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d89902d35888ede5953e4a06b412c89e0357b2e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.692ex; height:2.843ex;" alt="{\displaystyle 321=(13)(2)}"></span>) and two permutations with one cycle (<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 312=(132)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>312</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>132</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 312=(132)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de02ad48056cc3de7140c0cb247cfc3011349a2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.882ex; height:2.843ex;" alt="{\displaystyle 312=(132)}"></span> and <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 231=(123)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>231</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>123</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 231=(123)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c0f2e174ff525559f5680fe61abeac8524a9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.882ex; height:2.843ex;" alt="{\displaystyle 231=(123)}"></span>). Thus <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 \left[{3 \atop 3}\right]=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mn>3</mn> <mn>3</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{3 \atop 3}\right]=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7bfc9ad188ca0dfa714cd808f3562f076f26a38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.436ex; height:6.176ex;" alt="{\displaystyle \left[{3 \atop 3}\right]=1}"></span>, <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 \left[{3 \atop 2}\right]=3,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>3</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{3 \atop 2}\right]=3,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac6808f457658b70ba2bdef4e1793fb966402e58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.083ex; height:6.176ex;" alt="{\displaystyle \left[{3 \atop 2}\right]=3,}"></span> and <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 \left[{3 \atop 1}\right]=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mn>3</mn> <mn>1</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{3 \atop 1}\right]=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9351dd5e056eac91bda237dfa57c235d6a045e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.436ex; height:6.176ex;" alt="{\displaystyle \left[{3 \atop 1}\right]=2}"></span>. These can be seen to agree with the previous algebraic calculations of <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 s(n,k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(n,k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a462712cd2a4765947d2ca485db7b9d86cdc370f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.54ex; height:2.843ex;" alt="{\displaystyle s(n,k)}"></span> for <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=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5a5a42ced00df920fad4ab2d4acdb960a4105b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=3}"></span>.</p><figure typeof="mw:File/Thumb"><a href="/wiki/File:Stirling_number_of_the_first_kind_s(4,2).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Stirling_number_of_the_first_kind_s%284%2C2%29.svg/350px-Stirling_number_of_the_first_kind_s%284%2C2%29.svg.png" decoding="async" width="350" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Stirling_number_of_the_first_kind_s%284%2C2%29.svg/525px-Stirling_number_of_the_first_kind_s%284%2C2%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Stirling_number_of_the_first_kind_s%284%2C2%29.svg/700px-Stirling_number_of_the_first_kind_s%284%2C2%29.svg.png 2x" data-file-width="870" data-file-height="578" /></a><figcaption>s(4,2)=11</figcaption></figure> <p>For another example, the image at right shows that <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 \left[{4 \atop 2}\right]=11}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mn>4</mn> <mn>2</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>11</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{4 \atop 2}\right]=11}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/107cc5ba9e0aadd50da35b6f578edf25ab034801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.598ex; height:6.176ex;" alt="{\displaystyle \left[{4 \atop 2}\right]=11}"></span>: the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> on 4 objects has 3 permutations of the form </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 (\bullet \bullet )(\bullet \bullet )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>∙</mo> <mo>∙</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>∙</mo> <mo>∙</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\bullet \bullet )(\bullet \bullet )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15e2da9b9e9670ea11957cad880f4db644516bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.268ex; height:2.843ex;" alt="{\displaystyle (\bullet \bullet )(\bullet \bullet )}"></span> (having 2 orbits, each of size 2),</dd></dl> <p>and 8 permutations of the form </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 (\bullet \bullet \bullet )(\bullet )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>∙</mo> <mo>∙</mo> <mo>∙</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>∙</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\bullet \bullet \bullet )(\bullet )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/733f24520765fcea6a1c5c601eae7e8b1dad9694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.301ex; height:2.843ex;" alt="{\displaystyle (\bullet \bullet \bullet )(\bullet )}"></span> (having 1 orbit of size 3 and 1 orbit of size 1).</dd></dl> <p>These numbers can be calculated by considering the orbits as <a href="/wiki/Conjugacy_class#Properties" title="Conjugacy class">conjugacy classes</a>. <a href="/wiki/Alfr%C3%A9d_R%C3%A9nyi" title="Alfréd Rényi">Alfréd Rényi</a> observed that the unsigned Stirling number of the first kind <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 \left[{n \atop k}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{n \atop k}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e244be41a40ffd185e9edea632ba557f6ad945" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.147ex; height:4.843ex;" alt="{\displaystyle \left[{n \atop k}\right]}"></span> also counts the number of permutations of size <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> with <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> left-to-right maxima.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Signs">Signs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=4" title="Edit section: Signs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The signs of the signed Stirling numbers of the first kind depend only on the parity of <span class="texhtml"><i>n</i> − <i>k</i></span>. </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 s(n,k)=(-1)^{n-k}\left[{n \atop k}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(n,k)=(-1)^{n-k}\left[{n \atop k}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72dcc2f162b842434c5cc9b3ca9ec36d65cbe9f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.339ex; height:4.843ex;" alt="{\displaystyle s(n,k)=(-1)^{n-k}\left[{n \atop k}\right].}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Recurrence_relation">Recurrence relation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=5" title="Edit section: Recurrence relation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The unsigned Stirling numbers of the first kind follow the <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrence relation</a> </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 \left[{n+1 \atop k}\right]=n\left[{n \atop k}\right]+\left[{n \atop k-1}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mi>n</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{n+1 \atop k}\right]=n\left[{n \atop k}\right]+\left[{n \atop k-1}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/406f793535c38a92fa1ebfc14cb382a1e0be0fdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.504ex; height:6.176ex;" alt="{\displaystyle \left[{n+1 \atop k}\right]=n\left[{n \atop k}\right]+\left[{n \atop k-1}\right]}"></span></dd></dl> <p>for <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 k>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}"></span>, with the boundary conditions </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 \left[{0 \atop 0}\right]=1\quad {\mbox{and}}\quad \left[{n \atop 0}\right]=\left[{0 \atop n}\right]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mn>0</mn> <mn>0</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>and</mtext> </mstyle> </mrow> <mspace width="1em" /> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mn>0</mn> <mi>n</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{0 \atop 0}\right]=1\quad {\mbox{and}}\quad \left[{n \atop 0}\right]=\left[{0 \atop n}\right]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0dfde68d7c6385ed755c9361e98c406106fd975" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.129ex; height:6.176ex;" alt="{\displaystyle \left[{0 \atop 0}\right]=1\quad {\mbox{and}}\quad \left[{n \atop 0}\right]=\left[{0 \atop n}\right]=0}"></span></dd></dl> <p>for <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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}"></span>.<sup id="cite_ref-:1_2-1" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>It follows immediately that the signed Stirling numbers of the first kind satisfy the recurrence </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 s(n+1,k)=-n\cdot s(n,k)+s(n,k-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>n</mi> <mo>⋅</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(n+1,k)=-n\cdot s(n,k)+s(n,k-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/793e2448ea5b4304c37a30ffe95123b02230814f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.445ex; height:2.843ex;" alt="{\displaystyle s(n+1,k)=-n\cdot s(n,k)+s(n,k-1)}"></span>.</dd></dl> <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style=""><strong>Algebraic proof</strong> <p>We prove the recurrence relation using the definition of Stirling numbers in terms of rising factorials. Distributing the last term of the product, we have </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 x^{\overline {n+1}}=x(x+1)\cdots (x+n-1)(x+n)=n\cdot x^{\overline {n}}+x\cdot x^{\overline {n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo accent="false">¯</mo> </mover> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo accent="false">¯</mo> </mover> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\overline {n+1}}=x(x+1)\cdots (x+n-1)(x+n)=n\cdot x^{\overline {n}}+x\cdot x^{\overline {n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/765856fc5d9450976601a4fd221d89ff50220137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.512ex; height:3.676ex;" alt="{\displaystyle x^{\overline {n+1}}=x(x+1)\cdots (x+n-1)(x+n)=n\cdot x^{\overline {n}}+x\cdot x^{\overline {n}}.}"></span></dd></dl> <p>The coefficient of <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 x^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/525e1133440e1565055dec6243aaf0f27d4d4e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.418ex; height:2.676ex;" alt="{\displaystyle x^{k}}"></span> on the left-hand side of this equation is <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 \left[{n+1 \atop k}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{n+1 \atop k}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/008094efc35dee4fa67257635a721896560580a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.41ex; height:6.176ex;" alt="{\displaystyle \left[{n+1 \atop k}\right]}"></span>. The coefficient of <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 x^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/525e1133440e1565055dec6243aaf0f27d4d4e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.418ex; height:2.676ex;" alt="{\displaystyle x^{k}}"></span> in <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\cdot x^{\overline {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo accent="false">¯</mo> </mover> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\cdot x^{\overline {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ebb299954df69c1f5fa62f238b9db23bfd44f52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.715ex; height:2.843ex;" alt="{\displaystyle n\cdot x^{\overline {n}}}"></span> is <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\cdot \left[{n \atop k}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>⋅</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\cdot \left[{n \atop k}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1407a6abc7880d2bd0b3c8fd1da79a62824b80c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.221ex; height:4.843ex;" alt="{\displaystyle n\cdot \left[{n \atop k}\right]}"></span>, while the coefficient of <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 x^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/525e1133440e1565055dec6243aaf0f27d4d4e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.418ex; height:2.676ex;" alt="{\displaystyle x^{k}}"></span> in <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 x\cdot x^{\overline {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo accent="false">¯</mo> </mover> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot x^{\overline {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c730c9bfce9be744b6aae3834a7910e7859de9dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.65ex; height:2.843ex;" alt="{\displaystyle x\cdot x^{\overline {n}}}"></span> is <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 \left[{n \atop k-1}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{n \atop k-1}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8d3eae72b220a14276f8c4746321cb122e1f275" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.226ex; height:6.176ex;" alt="{\displaystyle \left[{n \atop k-1}\right]}"></span>. Since the two sides are equal as polynomials, the coefficients of <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 x^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/525e1133440e1565055dec6243aaf0f27d4d4e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.418ex; height:2.676ex;" alt="{\displaystyle x^{k}}"></span> on both sides must be equal, and the result follows. </p> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Combinatorial proof</strong> <p>We prove the recurrence relation using the definition of Stirling numbers in terms of permutations with a given number of cycles (or equivalently, <a href="/wiki/Orbit_(group_theory)" class="mw-redirect" title="Orbit (group theory)">orbits</a>). </p><p>Consider forming a permutation of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"></span> objects from a permutation of <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> objects by adding a distinguished object. There are exactly two ways in which this can be accomplished. We could do this by forming a <a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">singleton</a> cycle, i.e., leaving the extra object alone. This increases the number of cycles by 1 and so accounts for the <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 \left[{n \atop k-1}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{n \atop k-1}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8d3eae72b220a14276f8c4746321cb122e1f275" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.226ex; height:6.176ex;" alt="{\displaystyle \left[{n \atop k-1}\right]}"></span> term in the recurrence formula. We could also insert the new object into one of the existing cycles. Consider an arbitrary permutation of <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> objects with <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> cycles, and label the objects <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 a_{1},\dots ,a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\dots ,a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852381be25b656d697c7a4a9634d3dc4c182d833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.009ex;" alt="{\displaystyle a_{1},\dots ,a_{n}}"></span>, so that the permutation is represented by </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 \displaystyle \underbrace {(a_{1}\ldots a_{j_{1}})(a_{j_{1}+1}\ldots a_{j_{2}})\ldots (a_{j_{k-1}+1}\ldots a_{n})} _{k\ \mathrm {cycles} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> <mo>…</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </munder> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle \underbrace {(a_{1}\ldots a_{j_{1}})(a_{j_{1}+1}\ldots a_{j_{2}})\ldots (a_{j_{k-1}+1}\ldots a_{n})} _{k\ \mathrm {cycles} }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b73a1c2427d844c56ed473766a4580270d91a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:42.616ex; height:6.843ex;" alt="{\displaystyle \displaystyle \underbrace {(a_{1}\ldots a_{j_{1}})(a_{j_{1}+1}\ldots a_{j_{2}})\ldots (a_{j_{k-1}+1}\ldots a_{n})} _{k\ \mathrm {cycles} }.}"></span></dd></dl> <p>To form a new permutation of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"></span> objects and <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> cycles one must insert the new object into this array. There are <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> ways to perform this insertion, inserting the new object immediately following any of the <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 a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"></span> already present. This explains the <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\left[{n \atop k}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\left[{n \atop k}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f0ae96627020c5b20342d641138e945c80298e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.929ex; height:4.843ex;" alt="{\displaystyle n\left[{n \atop k}\right]}"></span> term of the recurrence relation. These two cases include all possibilities, so the recurrence relation follows. </p> </div> <div class="mw-heading mw-heading2"><h2 id="Table_of_values">Table of values</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=6" title="Edit section: Table of values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Below is a <a href="/wiki/Triangular_array" title="Triangular array">triangular array</a> of unsigned values for the Stirling numbers of the first kind, similar in form to <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>. These values are easy to generate using the recurrence relation in the previous section. </p> <table cellspacing="0" cellpadding="5" style="text-align:right;" class="wikitable"> <tbody><tr> <td 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"><i>k</i></div><div style="margin-right:2em;text-align:left"><b>n</b></div> </td> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">0 </th> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">1 </th> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">2 </th> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">3 </th> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">4 </th> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">5 </th> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">6 </th> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">7 </th> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">8 </th> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">9 </th> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">10 </th></tr> <tr> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">0 </th> <td>1 </td></tr> <tr> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">1 </th> <td>0 </td> <td>1 </td></tr> <tr> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">2 </th> <td>0 </td> <td>1 </td> <td>1 </td></tr> <tr> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">3 </th> <td>0 </td> <td>2 </td> <td>3 </td> <td>1 </td></tr> <tr> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">4 </th> <td>0 </td> <td>6 </td> <td>11 </td> <td>6 </td> <td>1 </td></tr> <tr> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">5 </th> <td>0 </td> <td>24 </td> <td>50 </td> <td>35 </td> <td>10 </td> <td>1 </td></tr> <tr> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">6 </th> <td>0 </td> <td>120 </td> <td>274 </td> <td>225 </td> <td>85 </td> <td>15 </td> <td>1 </td></tr> <tr> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">7 </th> <td>0 </td> <td>720 </td> <td>1764 </td> <td>1624 </td> <td>735 </td> <td>175 </td> <td>21 </td> <td>1 </td></tr> <tr> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">8 </th> <td>0 </td> <td>5040 </td> <td>13068 </td> <td>13132 </td> <td>6769 </td> <td>1960 </td> <td>322 </td> <td>28 </td> <td>1 </td></tr> <tr> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">9 </th> <td>0 </td> <td>40320 </td> <td>109584 </td> <td>118124 </td> <td>67284 </td> <td>22449 </td> <td>4536 </td> <td>546 </td> <td>36 </td> <td>1 </td></tr> <tr> <th style="background: #ececec; color: black; font-weight: bold; vertical-align: middle; text-align: right;" class="table-rh">10 </th> <td>0 </td> <td>362880 </td> <td>1026576 </td> <td>1172700 </td> <td>723680 </td> <td>269325 </td> <td>63273 </td> <td>9450 </td> <td>870 </td> <td>45 </td> <td>1 </td></tr> </tbody></table> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=7" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Simple_identities">Simple identities</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=8" title="Edit section: Simple identities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a> one has, </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 \left[{n \atop 0}\right]=\delta _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mn>0</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{n \atop 0}\right]=\delta _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4d51b87d7afa97bd6e9814b3414c50c7620bb40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.496ex; height:4.843ex;" alt="{\displaystyle \left[{n \atop 0}\right]=\delta _{n}}"></span></dd></dl> <p>and </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 \left[{0 \atop k}\right]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mn>0</mn> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{0 \atop k}\right]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f1e0a6054a5fa9f54c1cdbb4e31a735b918c5ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.485ex; height:6.176ex;" alt="{\displaystyle \left[{0 \atop k}\right]=0}"></span> if <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 k>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}"></span>, or more generally <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 \left[{n \atop k}\right]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{n \atop k}\right]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5a2fcff9da2ae84ea4dace07629dd21282b525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.408ex; height:4.843ex;" alt="{\displaystyle \left[{n \atop k}\right]=0}"></span> if <i>k</i> > <i>n</i>.</dd></dl> <p>Also </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 \left[{n \atop 1}\right]=(n-1)!,\quad \left[{n \atop n}\right]=1,\quad \left[{n \atop n-1}\right]={n \choose 2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mn>1</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>,</mo> <mspace width="1em" /> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>n</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em" /> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <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> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{n \atop 1}\right]=(n-1)!,\quad \left[{n \atop n}\right]=1,\quad \left[{n \atop n-1}\right]={n \choose 2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4d6ccfce18a86c6fe5ce0ebe29f6ce77122a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.191ex; height:6.176ex;" alt="{\displaystyle \left[{n \atop 1}\right]=(n-1)!,\quad \left[{n \atop n}\right]=1,\quad \left[{n \atop n-1}\right]={n \choose 2},}"></span></dd></dl> <p>and </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 \left[{n \atop n-2}\right]={\frac {3n-1}{4}}{n \choose 3}\quad {\mbox{ and }}\quad \left[{n \atop n-3}\right]={n \choose 2}{n \choose 4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>4</mn> </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"> <mi>n</mi> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> and </mtext> </mstyle> </mrow> <mspace width="1em" /> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <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> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <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> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{n \atop n-2}\right]={\frac {3n-1}{4}}{n \choose 3}\quad {\mbox{ and }}\quad \left[{n \atop n-3}\right]={n \choose 2}{n \choose 4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e5d4c56c9df7f88a1d0b29a49e0ffb6ecc8f9f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.449ex; height:6.176ex;" alt="{\displaystyle \left[{n \atop n-2}\right]={\frac {3n-1}{4}}{n \choose 3}\quad {\mbox{ and }}\quad \left[{n \atop n-3}\right]={n \choose 2}{n \choose 4}.}"></span></dd></dl> <p>Similar relationships involving the Stirling numbers hold for the <a href="/wiki/Bernoulli_polynomials" title="Bernoulli polynomials">Bernoulli polynomials</a>. Many relations for the Stirling numbers shadow similar relations on the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a>. The study of these 'shadow relationships' is termed <a href="/wiki/Umbral_calculus" title="Umbral calculus">umbral calculus</a> and culminates in the theory of <a href="/wiki/Sheffer_sequences" class="mw-redirect" title="Sheffer sequences">Sheffer sequences</a>. Generalizations of the <a href="/wiki/Stirling_numbers" class="mw-redirect" title="Stirling numbers">Stirling numbers</a> of both kinds to arbitrary complex-valued inputs may be extended through the relations of these triangles to the <a href="/wiki/Stirling_polynomial#Stirling_convolution_polynomials" class="mw-redirect" title="Stirling polynomial">Stirling convolution polynomials</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Combinatorial proofs</strong> <p>These identities may be derived by enumerating permutations directly. For example, a permutation of <i>n</i> elements with <i>n</i> − 3 cycles must have one of the following forms: </p> <ul><li><i>n</i> − 6 fixed points and three two-cycles</li> <li><i>n</i> − 5 fixed points, a three-cycle and a two-cycle, or</li> <li><i>n</i> − 4 fixed points and a four-cycle.</li></ul> <p>The three types may be enumerated as follows: </p> <ul><li>choose the six elements that go into the two-cycles, decompose them into two-cycles and take into account that the order of the cycles is not important:</li></ul> <dl><dd><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 \choose 6}{6 \choose 2,2,2}{\frac {1}{6}}}"> <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"> <mi>n</mi> <mn>6</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>6</mn> <mrow> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>2</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> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose 6}{6 \choose 2,2,2}{\frac {1}{6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa6aed0836f1838011bf9d985e0aa57bdefc6df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.791ex; height:6.176ex;" alt="{\displaystyle {n \choose 6}{6 \choose 2,2,2}{\frac {1}{6}}}"></span></dd></dl></dd></dl> <ul><li>choose the five elements that go into the three-cycle and the two-cycle, choose the elements of the three-cycle and take into account that three elements generate two three-cycles:</li></ul> <dl><dd><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 \choose 5}{5 \choose 3}\times 2}"> <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"> <mi>n</mi> <mn>5</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>5</mn> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose 5}{5 \choose 3}\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c4d4a96cef52580dff5504ffe64eb9116107ff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.402ex; height:6.176ex;" alt="{\displaystyle {n \choose 5}{5 \choose 3}\times 2}"></span></dd></dl></dd></dl> <ul><li>choose the four elements of the four-cycle and take into account that four elements generate six four-cycles:</li></ul> <dl><dd><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 \choose 4}\times 6.}"> <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"> <mi>n</mi> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>×</mo> <mn>6.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose 4}\times 6.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0811add49952edb986a1b925720f03afc9356e66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.466ex; height:6.176ex;" alt="{\displaystyle {n \choose 4}\times 6.}"></span></dd></dl></dd></dl> <p>Sum the three contributions to obtain </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 \choose 6}{6 \choose 2,2,2}{\frac {1}{6}}+{n \choose 5}{5 \choose 3}\times 2+{n \choose 4}\times 6={n \choose 2}{n \choose 4}.}"> <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"> <mi>n</mi> <mn>6</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>6</mn> <mrow> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>2</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> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>+</mo> <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> <mn>5</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>5</mn> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>×</mo> <mn>2</mn> <mo>+</mo> <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> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>×</mo> <mn>6</mn> <mo>=</mo> <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> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <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> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose 6}{6 \choose 2,2,2}{\frac {1}{6}}+{n \choose 5}{5 \choose 3}\times 2+{n \choose 4}\times 6={n \choose 2}{n \choose 4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c84ca8581ea402051f23c0393eea45c3941128df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.07ex; height:6.176ex;" alt="{\displaystyle {n \choose 6}{6 \choose 2,2,2}{\frac {1}{6}}+{n \choose 5}{5 \choose 3}\times 2+{n \choose 4}\times 6={n \choose 2}{n \choose 4}.}"></span></dd></dl> </div> <p>Note that all the combinatorial proofs above use either binomials or multinomials of <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>. </p><p>Therefore if <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is prime, then: </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 p\ |\left[{p \atop k}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>p</mi> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\ |\left[{p \atop k}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c396b066ef675ce81afa21ba539df0828627ceb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:6.837ex; height:4.843ex;" alt="{\displaystyle p\ |\left[{p \atop k}\right]}"></span> for <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<k<p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1<k<p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b20a6fa23bf9ef059f503d3df77cbaf8cfc78875" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.74ex; height:2.509ex;" alt="{\displaystyle 1<k<p}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Expansions_for_fixed_k">Expansions for fixed <i>k</i></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=9" title="Edit section: Expansions for fixed k"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since the Stirling numbers are the coefficients of a polynomial with roots 0, 1, ..., <span class="texhtml"><i>n</i> − 1</span>, one has by <a href="/wiki/Vieta%27s_formulas" title="Vieta's formulas">Vieta's formulas</a> that </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{matrix}n\\n-k\end{matrix}}\right]=\sum _{0\leq i_{1}<\ldots <i_{k}<n}i_{1}i_{2}\cdots i_{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>≤</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>…</mo> <mo><</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo><</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{matrix}n\\n-k\end{matrix}}\right]=\sum _{0\leq i_{1}<\ldots <i_{k}<n}i_{1}i_{2}\cdots i_{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef2b9f629d23028e10d7f9772e72c9066339fc6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.561ex; height:7.009ex;" alt="{\displaystyle \left[{\begin{matrix}n\\n-k\end{matrix}}\right]=\sum _{0\leq i_{1}<\ldots <i_{k}<n}i_{1}i_{2}\cdots i_{k}.}"></span> </p><p>In other words, the Stirling numbers of the first kind are given by <a href="/wiki/Elementary_symmetric_polynomials" class="mw-redirect" title="Elementary symmetric polynomials">elementary symmetric polynomials</a> evaluated at 0, 1, ..., <span class="texhtml"><i>n</i> − 1</span>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In this form, the simple identities given above take the form </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{matrix}n\\n-1\end{matrix}}\right]=\sum _{i=0}^{n-1}i={\binom {n}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <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> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>i</mi> <mo>=</mo> <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> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{matrix}n\\n-1\end{matrix}}\right]=\sum _{i=0}^{n-1}i={\binom {n}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/274f8796cc3eb540639bd74154676caee7efad96" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.808ex; height:7.343ex;" alt="{\displaystyle \left[{\begin{matrix}n\\n-1\end{matrix}}\right]=\sum _{i=0}^{n-1}i={\binom {n}{2}},}"></span><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{matrix}n\\n-2\end{matrix}}\right]=\sum _{i=0}^{n-1}\sum _{j=0}^{i-1}ij={\frac {3n-1}{4}}{\binom {n}{3}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <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> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>i</mi> <mi>j</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>4</mn> </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"> <mi>n</mi> <mn>3</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{matrix}n\\n-2\end{matrix}}\right]=\sum _{i=0}^{n-1}\sum _{j=0}^{i-1}ij={\frac {3n-1}{4}}{\binom {n}{3}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34cc76556b680ce3b4e39f67adec4e93426a95f5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.905ex; height:7.676ex;" alt="{\displaystyle \left[{\begin{matrix}n\\n-2\end{matrix}}\right]=\sum _{i=0}^{n-1}\sum _{j=0}^{i-1}ij={\frac {3n-1}{4}}{\binom {n}{3}},}"></span><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{matrix}n\\n-3\end{matrix}}\right]=\sum _{i=0}^{n-1}\sum _{j=0}^{i-1}\sum _{k=0}^{j-1}ijk={\binom {n}{2}}{\binom {n}{4}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <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> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mo>=</mo> <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> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <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> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{matrix}n\\n-3\end{matrix}}\right]=\sum _{i=0}^{n-1}\sum _{j=0}^{i-1}\sum _{k=0}^{j-1}ijk={\binom {n}{2}}{\binom {n}{4}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b77a3abb9cfd6556e93da99c1c0cb92be143d6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:39.278ex; height:7.676ex;" alt="{\displaystyle \left[{\begin{matrix}n\\n-3\end{matrix}}\right]=\sum _{i=0}^{n-1}\sum _{j=0}^{i-1}\sum _{k=0}^{j-1}ijk={\binom {n}{2}}{\binom {n}{4}},}"></span> and so on. </p><p>One may produce alternative forms for the Stirling numbers of the first kind with a similar approach preceded by some algebraic manipulation: since </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 (x+1)(x+2)\cdots (x+n-1)=(n-1)!\cdot (x+1)\left({\frac {x}{2}}+1\right)\cdots \left({\frac {x}{n-1}}+1\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+1)(x+2)\cdots (x+n-1)=(n-1)!\cdot (x+1)\left({\frac {x}{2}}+1\right)\cdots \left({\frac {x}{n-1}}+1\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9927e336c7f40141ab3f40f6c55dbaf91a854ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:76.452ex; height:6.176ex;" alt="{\displaystyle (x+1)(x+2)\cdots (x+n-1)=(n-1)!\cdot (x+1)\left({\frac {x}{2}}+1\right)\cdots \left({\frac {x}{n-1}}+1\right),}"></span></dd></dl> <p>it follows from <a href="/wiki/Newton%27s_identities" title="Newton's identities">Newton's formulas</a> that one can expand the Stirling numbers of the first kind in terms of <a href="/wiki/Generalized_harmonic_number" class="mw-redirect" title="Generalized harmonic number">generalized harmonic numbers</a>. This yields identities like </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 \left[{n \atop 2}\right]=(n-1)!\;H_{n-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mspace width="thickmathspace" /> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{n \atop 2}\right]=(n-1)!\;H_{n-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/096c7e40db02f5964d457ec80219863546634605" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.641ex; height:4.843ex;" alt="{\displaystyle \left[{n \atop 2}\right]=(n-1)!\;H_{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 \left[{n \atop 3}\right]={\frac {1}{2}}(n-1)!\left[(H_{n-1})^{2}-H_{n-1}^{(2)}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mn>3</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{n \atop 3}\right]={\frac {1}{2}}(n-1)!\left[(H_{n-1})^{2}-H_{n-1}^{(2)}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4cb2bd8a2c91ee0fc310c9dcd26cca6ff5c56de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.884ex; height:5.176ex;" alt="{\displaystyle \left[{n \atop 3}\right]={\frac {1}{2}}(n-1)!\left[(H_{n-1})^{2}-H_{n-1}^{(2)}\right]}"></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 \left[{n \atop 4}\right]={\frac {1}{3!}}(n-1)!\left[(H_{n-1})^{3}-3H_{n-1}H_{n-1}^{(2)}+2H_{n-1}^{(3)}\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mn>4</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{n \atop 4}\right]={\frac {1}{3!}}(n-1)!\left[(H_{n-1})^{3}-3H_{n-1}H_{n-1}^{(2)}+2H_{n-1}^{(3)}\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ba7d7fde482df6c5226061af71864901e88c4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:53.231ex; height:5.343ex;" alt="{\displaystyle \left[{n \atop 4}\right]={\frac {1}{3!}}(n-1)!\left[(H_{n-1})^{3}-3H_{n-1}H_{n-1}^{(2)}+2H_{n-1}^{(3)}\right],}"></span></dd></dl> <p>where <i>H</i><sub><i>n</i></sub> is the <a href="/wiki/Harmonic_number" title="Harmonic number">harmonic number</a> <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 H_{n}={\frac {1}{1}}+{\frac {1}{2}}+\ldots +{\frac {1}{n}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}={\frac {1}{1}}+{\frac {1}{2}}+\ldots +{\frac {1}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62dfc8e2ce87d2123048d220749d3c81d14381a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.721ex; height:5.176ex;" alt="{\displaystyle H_{n}={\frac {1}{1}}+{\frac {1}{2}}+\ldots +{\frac {1}{n}}}"></span> and <i>H</i><sub><i>n</i></sub><sup>(<i>m</i>)</sup> is the generalized harmonic number <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}^{(m)}={\frac {1}{1^{m}}}+{\frac {1}{2^{m}}}+\ldots +{\frac {1}{n^{m}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}^{(m)}={\frac {1}{1^{m}}}+{\frac {1}{2^{m}}}+\ldots +{\frac {1}{n^{m}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fdf9e372ee406c038f6803c8df956b01b72c44a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.301ex; height:5.343ex;" alt="{\displaystyle H_{n}^{(m)}={\frac {1}{1^{m}}}+{\frac {1}{2^{m}}}+\ldots +{\frac {1}{n^{m}}}.}"></span> </p><p>These relations can be generalized to give </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 {\frac {1}{(n-1)!}}\left[{\begin{matrix}n\\k+1\end{matrix}}\right]=\sum _{i_{1}=1}^{n-1}\sum _{i_{2}=i_{1}+1}^{n-1}\cdots \sum _{i_{k}=i_{k-1}+1}^{n-1}{\frac {1}{i_{1}i_{2}\cdots i_{k}}}={\frac {w(n,k)}{k!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo>⋯</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <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>1</mn> <mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>w</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(n-1)!}}\left[{\begin{matrix}n\\k+1\end{matrix}}\right]=\sum _{i_{1}=1}^{n-1}\sum _{i_{2}=i_{1}+1}^{n-1}\cdots \sum _{i_{k}=i_{k-1}+1}^{n-1}{\frac {1}{i_{1}i_{2}\cdots i_{k}}}={\frac {w(n,k)}{k!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9a06fafe850fa6dd3725a4fa69df45eec6dad01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:63.469ex; height:7.843ex;" alt="{\displaystyle {\frac {1}{(n-1)!}}\left[{\begin{matrix}n\\k+1\end{matrix}}\right]=\sum _{i_{1}=1}^{n-1}\sum _{i_{2}=i_{1}+1}^{n-1}\cdots \sum _{i_{k}=i_{k-1}+1}^{n-1}{\frac {1}{i_{1}i_{2}\cdots i_{k}}}={\frac {w(n,k)}{k!}}}"></span></dd></dl> <p>where <i>w</i>(<i>n</i>, <i>m</i>) is defined recursively in terms of the generalized harmonic numbers by </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 w(n,m)=\delta _{m,0}+\sum _{k=0}^{m-1}(1-m)_{k}H_{n-1}^{(k+1)}w(n,m-1-k).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <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> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>w</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w(n,m)=\delta _{m,0}+\sum _{k=0}^{m-1}(1-m)_{k}H_{n-1}^{(k+1)}w(n,m-1-k).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e808312a92449614e2d602eadd0d05568bf7782" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.569ex; height:7.509ex;" alt="{\displaystyle w(n,m)=\delta _{m,0}+\sum _{k=0}^{m-1}(1-m)_{k}H_{n-1}^{(k+1)}w(n,m-1-k).}"></span></dd></dl> <p>(Here <i>δ</i> is the <a href="/wiki/Kronecker_delta_function" class="mw-redirect" title="Kronecker delta function">Kronecker delta function</a> and <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 (m)_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>m</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (m)_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee82e44af7d5181dac8109dce12a2af022e95722" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.938ex; height:2.843ex;" alt="{\displaystyle (m)_{k}}"></span> is the <a href="/wiki/Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">Pochhammer symbol</a>.)<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>For fixed <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\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}"></span> these weighted harmonic number expansions are generated by the generating function </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 {\frac {1}{n!}}\left[{\begin{matrix}n+1\\k\end{matrix}}\right]=[x^{k}]\exp \left(\sum _{m\geq 1}{\frac {(-1)^{m-1}H_{n}^{(m)}}{m}}x^{m}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>≥</mo> <mn>1</mn> </mrow> </munder> <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>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mi>m</mi> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n!}}\left[{\begin{matrix}n+1\\k\end{matrix}}\right]=[x^{k}]\exp \left(\sum _{m\geq 1}{\frac {(-1)^{m-1}H_{n}^{(m)}}{m}}x^{m}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0556e9fee4806000845199f06e178a4f82eb66f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.332ex; height:7.676ex;" alt="{\displaystyle {\frac {1}{n!}}\left[{\begin{matrix}n+1\\k\end{matrix}}\right]=[x^{k}]\exp \left(\sum _{m\geq 1}{\frac {(-1)^{m-1}H_{n}^{(m)}}{m}}x^{m}\right),}"></span></dd></dl> <p>where the notation <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 [x^{k}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x^{k}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05c7a1f1d2594bf84cfe3bf00686732c5489b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.712ex; height:3.176ex;" alt="{\displaystyle [x^{k}]}"></span> means extraction of the coefficient of <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 x^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/525e1133440e1565055dec6243aaf0f27d4d4e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.418ex; height:2.676ex;" alt="{\displaystyle x^{k}}"></span> from the following <a href="/wiki/Formal_power_series" title="Formal power series">formal power series</a> (see the non-exponential <a href="/wiki/Bell_polynomials" title="Bell polynomials">Bell polynomials</a> and section 3 of <sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>). </p><p>More generally, sums related to these weighted harmonic number expansions of the Stirling numbers of the first kind can be defined through generalized zeta series <a href="/wiki/Generating_function_transformation#Derivative_transformations" title="Generating function transformation">transforms of generating functions</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>One can also "invert" the relations for these Stirling numbers given in terms of the <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>-order harmonic numbers to write the integer-order generalized harmonic numbers in terms of weighted sums of terms involving the Stirling numbers of the first kind. For example, when <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 k=2,3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=2,3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efe98c233aa59b8ca89c393123c56df76d6b91c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.669ex; height:2.509ex;" alt="{\displaystyle k=2,3}"></span> the second-order and third-order harmonic numbers are given by </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}\cdot H_{n}^{(2)}=\left[{\begin{matrix}n+1\\2\end{matrix}}\right]^{2}-2\left[{\begin{matrix}n+1\\1\end{matrix}}\right]\left[{\begin{matrix}n+1\\3\end{matrix}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n!)^{2}\cdot H_{n}^{(2)}=\left[{\begin{matrix}n+1\\2\end{matrix}}\right]^{2}-2\left[{\begin{matrix}n+1\\1\end{matrix}}\right]\left[{\begin{matrix}n+1\\3\end{matrix}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b31ec37dee337d75e7bbfe72373d44056bbb3e2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.763ex; height:6.509ex;" alt="{\displaystyle (n!)^{2}\cdot H_{n}^{(2)}=\left[{\begin{matrix}n+1\\2\end{matrix}}\right]^{2}-2\left[{\begin{matrix}n+1\\1\end{matrix}}\right]\left[{\begin{matrix}n+1\\3\end{matrix}}\right]}"></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 (n!)^{3}\cdot H_{n}^{(3)}=\left[{\begin{matrix}n+1\\2\end{matrix}}\right]^{3}-3\left[{\begin{matrix}n+1\\1\end{matrix}}\right]\left[{\begin{matrix}n+1\\2\end{matrix}}\right]\left[{\begin{matrix}n+1\\3\end{matrix}}\right]+3\left[{\begin{matrix}n+1\\1\end{matrix}}\right]^{2}\left[{\begin{matrix}n+1\\4\end{matrix}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mn>3</mn> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n!)^{3}\cdot H_{n}^{(3)}=\left[{\begin{matrix}n+1\\2\end{matrix}}\right]^{3}-3\left[{\begin{matrix}n+1\\1\end{matrix}}\right]\left[{\begin{matrix}n+1\\2\end{matrix}}\right]\left[{\begin{matrix}n+1\\3\end{matrix}}\right]+3\left[{\begin{matrix}n+1\\1\end{matrix}}\right]^{2}\left[{\begin{matrix}n+1\\4\end{matrix}}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bbe74648bd6cff66777abd5f9ffb08c375184b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:78.44ex; height:6.509ex;" alt="{\displaystyle (n!)^{3}\cdot H_{n}^{(3)}=\left[{\begin{matrix}n+1\\2\end{matrix}}\right]^{3}-3\left[{\begin{matrix}n+1\\1\end{matrix}}\right]\left[{\begin{matrix}n+1\\2\end{matrix}}\right]\left[{\begin{matrix}n+1\\3\end{matrix}}\right]+3\left[{\begin{matrix}n+1\\1\end{matrix}}\right]^{2}\left[{\begin{matrix}n+1\\4\end{matrix}}\right].}"></span></dd></dl> <p>More generally, one can invert the <a href="/wiki/Bell_polynomial" class="mw-redirect" title="Bell polynomial">Bell polynomial</a> generating function for the Stirling numbers expanded in terms of the <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>-order <a href="/wiki/Harmonic_number" title="Harmonic number">harmonic numbers</a> to obtain that for integers <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 m\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\geq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca2e437e89ef4565a87f1a6d90ed37eef1d8ce3" 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 2}"></span> </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 H_{n}^{(m)}=-m\times [x^{m}]\log \left(1+\sum _{k\geq 1}\left[{\begin{matrix}n+1\\k+1\end{matrix}}\right]{\frac {(-x)^{k}}{n!}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mi>m</mi> <mo>×</mo> <mo stretchy="false">[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </munder> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}^{(m)}=-m\times [x^{m}]\log \left(1+\sum _{k\geq 1}\left[{\begin{matrix}n+1\\k+1\end{matrix}}\right]{\frac {(-x)^{k}}{n!}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d9abfcda5a2fd80bd18d96f81bf6cc6740ec56b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:50.439ex; height:7.676ex;" alt="{\displaystyle H_{n}^{(m)}=-m\times [x^{m}]\log \left(1+\sum _{k\geq 1}\left[{\begin{matrix}n+1\\k+1\end{matrix}}\right]{\frac {(-x)^{k}}{n!}}\right).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Finite_sums">Finite sums</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=10" title="Edit section: Finite sums"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since permutations are partitioned by number of cycles, one has </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 \sum _{k=0}^{n}\left[{n \atop k}\right]=n!}"> <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> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{n}\left[{n \atop k}\right]=n!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b3d2ba78741f8131184b255a1b4b746c6e639a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.029ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{n}\left[{n \atop k}\right]=n!}"></span></dd></dl> <p>The identities </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 \sum _{k=0}^{n}\left[{n \atop k}\right]u^{k}=n!{\binom {n+u-1}{u-1}},\,u>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> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mi>n</mi> <mo>!</mo> <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> <mi>u</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>u</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>u</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{n}\left[{n \atop k}\right]u^{k}=n!{\binom {n+u-1}{u-1}},\,u>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17427aaf6f35b354bc540bf04cb1c26985e3310f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.835ex; height:7.009ex;" alt="{\displaystyle \sum _{k=0}^{n}\left[{n \atop k}\right]u^{k}=n!{\binom {n+u-1}{u-1}},\,u>0}"></span></dd></dl> <p>and </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 \sum _{p=k}^{n}{\left[{n \atop p}\right]{\binom {p}{k}}}=\left[{n+1 \atop k+1}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>p</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>p</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{p=k}^{n}{\left[{n \atop p}\right]{\binom {p}{k}}}=\left[{n+1 \atop k+1}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d2d97089fccbb78bd20645cddc0dc608d0aa5fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.677ex; height:7.176ex;" alt="{\displaystyle \sum _{p=k}^{n}{\left[{n \atop p}\right]{\binom {p}{k}}}=\left[{n+1 \atop k+1}\right]}"></span></dd></dl> <p>can be proved by the techniques at <a href="/wiki/Stirling_numbers_and_exponential_generating_functions#Stirling_numbers_of_the_first_kind" class="mw-redirect" title="Stirling numbers and exponential generating functions">Stirling numbers and exponential generating functions#Stirling numbers of the first kind</a> and <a href="/wiki/Binomial_coefficient#Ordinary_generating_functions" title="Binomial coefficient">Binomial coefficient#Ordinary generating functions</a>. </p><p>The table in section 6.1 of <i>Concrete Mathematics</i> provides a plethora of generalized forms of finite sums involving the Stirling numbers. Several particular finite sums relevant to this article include </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 {\begin{aligned}\left[{n \atop m}\right]&=\sum _{k=m}^{n}\left[{n+1 \atop k+1}\right]{\binom {k}{m}}(-1)^{m-k}\\\left[{n+1 \atop m+1}\right]&=\sum _{k=m}^{n}\left[{k \atop m}\right]{\frac {n!}{k!}}\\\left[{m+n+1 \atop m}\right]&=\sum _{k=0}^{m}(n+k)\left[{n+k \atop k}\right]\\\left[{n \atop l+m}\right]{\binom {l+m}{l}}&=\sum _{k}\left[{k \atop l}\right]\left[{n-k \atop m}\right]{\binom {n}{k}}.\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> <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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <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> <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>m</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>m</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <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> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>k</mi> <mi>m</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>m</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </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> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mi>l</mi> <mo>+</mo> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </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>l</mi> <mo>+</mo> <mi>m</mi> </mrow> <mi>l</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>k</mi> <mi>l</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> <mi>m</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <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> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left[{n \atop m}\right]&=\sum _{k=m}^{n}\left[{n+1 \atop k+1}\right]{\binom {k}{m}}(-1)^{m-k}\\\left[{n+1 \atop m+1}\right]&=\sum _{k=m}^{n}\left[{k \atop m}\right]{\frac {n!}{k!}}\\\left[{m+n+1 \atop m}\right]&=\sum _{k=0}^{m}(n+k)\left[{n+k \atop k}\right]\\\left[{n \atop l+m}\right]{\binom {l+m}{l}}&=\sum _{k}\left[{k \atop l}\right]\left[{n-k \atop m}\right]{\binom {n}{k}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a529ab7cff6295420342dae0ad74e051f411d96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.266ex; margin-bottom: -0.239ex; width:48.632ex; height:28.176ex;" alt="{\displaystyle {\begin{aligned}\left[{n \atop m}\right]&=\sum _{k=m}^{n}\left[{n+1 \atop k+1}\right]{\binom {k}{m}}(-1)^{m-k}\\\left[{n+1 \atop m+1}\right]&=\sum _{k=m}^{n}\left[{k \atop m}\right]{\frac {n!}{k!}}\\\left[{m+n+1 \atop m}\right]&=\sum _{k=0}^{m}(n+k)\left[{n+k \atop k}\right]\\\left[{n \atop l+m}\right]{\binom {l+m}{l}}&=\sum _{k}\left[{k \atop l}\right]\left[{n-k \atop m}\right]{\binom {n}{k}}.\end{aligned}}}"></span></dd></dl> <p>Additionally, if we define the <a href="/wiki/Eulerian_numbers#Eulerian_numbers_of_the_second_kind" class="mw-redirect" title="Eulerian numbers"><i>second-order</i> Eulerian numbers</a> by the triangular recurrence relation <sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </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 \left\langle \!\!\left\langle {n \atop k}\right\rangle \!\!\right\rangle =(k+1)\left\langle \!\!\left\langle {n-1 \atop k}\right\rangle \!\!\right\rangle +(2n-1-k)\left\langle \!\!\left\langle {n-1 \atop k-1}\right\rangle \!\!\right\rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mo>⟨</mo> <mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mo>⟩</mo> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow> <mo>⟨</mo> <mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <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> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mo>⟩</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle \!\!\left\langle {n \atop k}\right\rangle \!\!\right\rangle =(k+1)\left\langle \!\!\left\langle {n-1 \atop k}\right\rangle \!\!\right\rangle +(2n-1-k)\left\langle \!\!\left\langle {n-1 \atop k-1}\right\rangle \!\!\right\rangle ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e71cf60cd982112ed37a31a00fc6fdfce67a47a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.034ex; height:6.176ex;" alt="{\displaystyle \left\langle \!\!\left\langle {n \atop k}\right\rangle \!\!\right\rangle =(k+1)\left\langle \!\!\left\langle {n-1 \atop k}\right\rangle \!\!\right\rangle +(2n-1-k)\left\langle \!\!\left\langle {n-1 \atop k-1}\right\rangle \!\!\right\rangle ,}"></span></dd></dl> <p>we arrive at the following identity related to the form of the <a href="/wiki/Stirling_polynomial#Stirling_convolution_polynomials" class="mw-redirect" title="Stirling polynomial">Stirling convolution polynomials</a> which can be employed to generalize both Stirling number triangles to arbitrary real, or complex-valued, values of the input <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>: </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 \left[{x \atop x-n}\right]=\sum _{k=0}^{n}\left\langle \!\!\left\langle {n \atop k}\right\rangle \!\!\right\rangle {\binom {x+k}{2n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>x</mi> <mrow> <mi>x</mi> <mo>−</mo> <mi>n</mi> </mrow> </mfrac> </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> </mrow> </munderover> <mrow> <mo>⟨</mo> <mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mo>⟩</mo> </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>x</mi> <mo>+</mo> <mi>k</mi> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{x \atop x-n}\right]=\sum _{k=0}^{n}\left\langle \!\!\left\langle {n \atop k}\right\rangle \!\!\right\rangle {\binom {x+k}{2n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de2da43de8a6d4ed4136ef8d702bbe61e122fb2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.339ex; height:7.009ex;" alt="{\displaystyle \left[{x \atop x-n}\right]=\sum _{k=0}^{n}\left\langle \!\!\left\langle {n \atop k}\right\rangle \!\!\right\rangle {\binom {x+k}{2n}}.}"></span></dd></dl> <p>Particular expansions of the previous identity lead to the following identities expanding the Stirling numbers of the first kind for the first few small values of <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:=1,2,3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>:=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n:=1,2,3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f1f1a955120bbac74267070fad3e2b02f92e399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.695ex; height:2.509ex;" alt="{\displaystyle n:=1,2,3}"></span>: </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 {\begin{aligned}\left[{\begin{matrix}x\\x-1\end{matrix}}\right]&={\binom {x}{2}}\\\left[{\begin{matrix}x\\x-2\end{matrix}}\right]&={\binom {x}{4}}+2{\binom {x+1}{4}}\\\left[{\begin{matrix}x\\x-3\end{matrix}}\right]&={\binom {x}{6}}+8{\binom {x+1}{6}}+6{\binom {x+2}{6}}.\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> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>x</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>x</mi> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mn>2</mn> <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>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> <mo>−</mo> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>x</mi> <mn>6</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mn>8</mn> <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>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>6</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mn>6</mn> <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>x</mi> <mo>+</mo> <mn>2</mn> </mrow> <mn>6</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left[{\begin{matrix}x\\x-1\end{matrix}}\right]&={\binom {x}{2}}\\\left[{\begin{matrix}x\\x-2\end{matrix}}\right]&={\binom {x}{4}}+2{\binom {x+1}{4}}\\\left[{\begin{matrix}x\\x-3\end{matrix}}\right]&={\binom {x}{6}}+8{\binom {x+1}{6}}+6{\binom {x+2}{6}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ef6dae010b253340d2fc128ab5be5d6a3ddf2fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:43.3ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}\left[{\begin{matrix}x\\x-1\end{matrix}}\right]&={\binom {x}{2}}\\\left[{\begin{matrix}x\\x-2\end{matrix}}\right]&={\binom {x}{4}}+2{\binom {x+1}{4}}\\\left[{\begin{matrix}x\\x-3\end{matrix}}\right]&={\binom {x}{6}}+8{\binom {x+1}{6}}+6{\binom {x+2}{6}}.\end{aligned}}}"></span></dd></dl> <p>Software tools for working with finite sums involving <a href="/wiki/Stirling_numbers" class="mw-redirect" title="Stirling numbers">Stirling numbers</a> and <a href="/wiki/Eulerian_numbers" class="mw-redirect" title="Eulerian numbers">Eulerian numbers</a> are provided by the <a rel="nofollow" class="external text" href="http://www.risc.jku.at/research/combinat/software/ergosum/RISC/Stirling.html">RISC Stirling.m package</a> utilities in <i>Mathematica</i>. Other software packages for <i>guessing</i> formulas for sequences (and polynomial sequence sums) involving Stirling numbers and other special triangles is available for both <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> and <a href="/wiki/SageMath" title="SageMath">Sage</a> <a rel="nofollow" class="external text" href="https://github.com/maxieds/GuessPolynomialSequences">here</a> and <a rel="nofollow" class="external text" href="https://github.com/maxieds/sage-guess">here</a>, respectively.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Congruences">Congruences</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=11" title="Edit section: Congruences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following congruence identity may be proved via a <a href="/wiki/Generating_function" title="Generating function">generating function</a>-based approach:<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </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 {\begin{aligned}\left[{\begin{matrix}n\\m\end{matrix}}\right]&\equiv {\binom {\lfloor n/2\rfloor }{m-\lceil n/2\rceil }}=[x^{m}]\left(x^{\lceil n/2\rceil }(x+1)^{\lfloor n/2\rfloor }\right)&&{\pmod {2}},\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> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌋</mo> </mrow> <mrow> <mi>m</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌈</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌉</mo> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌈</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌉</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌋</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left[{\begin{matrix}n\\m\end{matrix}}\right]&\equiv {\binom {\lfloor n/2\rfloor }{m-\lceil n/2\rceil }}=[x^{m}]\left(x^{\lceil n/2\rceil }(x+1)^{\lfloor n/2\rfloor }\right)&&{\pmod {2}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9697227f05c85704f30e7bfef9ea45b43a6a0506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:66.998ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\left[{\begin{matrix}n\\m\end{matrix}}\right]&\equiv {\binom {\lfloor n/2\rfloor }{m-\lceil n/2\rceil }}=[x^{m}]\left(x^{\lceil n/2\rceil }(x+1)^{\lfloor n/2\rfloor }\right)&&{\pmod {2}},\end{aligned}}}"></span></dd></dl> <p>More recent results providing <a href="/wiki/Generating_function#Representation_by_continued_fractions_.28Jacobi-type_J-fractions.29" title="Generating function">Jacobi-type J-fractions</a> that generate the <a href="/wiki/Factorial" title="Factorial">single factorial function</a> and <a href="/wiki/Pochhammer_k-symbol" title="Pochhammer k-symbol">generalized factorial-related products</a> lead to other new congruence results for the Stirling numbers of the first kind.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> For example, working modulo <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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> we can prove that </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 {\begin{aligned}\left[{\begin{matrix}n\\1\end{matrix}}\right]&\equiv {\frac {2^{n}}{4}}[n\geq 2]+[n=1]&&{\pmod {2}}\\\left[{\begin{matrix}n\\2\end{matrix}}\right]&\equiv {\frac {3\cdot 2^{n}}{16}}(n-1)[n\geq 3]+[n=2]&&{\pmod {2}}\\\left[{\begin{matrix}n\\3\end{matrix}}\right]&\equiv 2^{n-7}(9n-20)(n-1)[n\geq 4]+[n=3]&&{\pmod {2}}\\\left[{\begin{matrix}n\\4\end{matrix}}\right]&\equiv 2^{n-9}(3n-10)(3n-7)(n-1)[n\geq 5]+[n=4]&&{\pmod {2}}\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> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>n</mi> <mo>≥</mo> <mn>2</mn> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mn>16</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo>≥</mo> <mn>3</mn> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">]</mo> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </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>7</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>9</mn> <mi>n</mi> <mo>−</mo> <mn>20</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo>≥</mo> <mn>4</mn> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo>=</mo> <mn>3</mn> <mo stretchy="false">]</mo> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> </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>9</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>3</mn> <mi>n</mi> <mo>−</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>n</mi> <mo>−</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo>≥</mo> <mn>5</mn> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo>=</mo> <mn>4</mn> <mo stretchy="false">]</mo> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left[{\begin{matrix}n\\1\end{matrix}}\right]&\equiv {\frac {2^{n}}{4}}[n\geq 2]+[n=1]&&{\pmod {2}}\\\left[{\begin{matrix}n\\2\end{matrix}}\right]&\equiv {\frac {3\cdot 2^{n}}{16}}(n-1)[n\geq 3]+[n=2]&&{\pmod {2}}\\\left[{\begin{matrix}n\\3\end{matrix}}\right]&\equiv 2^{n-7}(9n-20)(n-1)[n\geq 4]+[n=3]&&{\pmod {2}}\\\left[{\begin{matrix}n\\4\end{matrix}}\right]&\equiv 2^{n-9}(3n-10)(3n-7)(n-1)[n\geq 5]+[n=4]&&{\pmod {2}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3500bd7d5e34c868f16b0434890ec3e32ee39df2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.838ex; width:70.271ex; height:24.843ex;" alt="{\displaystyle {\begin{aligned}\left[{\begin{matrix}n\\1\end{matrix}}\right]&\equiv {\frac {2^{n}}{4}}[n\geq 2]+[n=1]&&{\pmod {2}}\\\left[{\begin{matrix}n\\2\end{matrix}}\right]&\equiv {\frac {3\cdot 2^{n}}{16}}(n-1)[n\geq 3]+[n=2]&&{\pmod {2}}\\\left[{\begin{matrix}n\\3\end{matrix}}\right]&\equiv 2^{n-7}(9n-20)(n-1)[n\geq 4]+[n=3]&&{\pmod {2}}\\\left[{\begin{matrix}n\\4\end{matrix}}\right]&\equiv 2^{n-9}(3n-10)(3n-7)(n-1)[n\geq 5]+[n=4]&&{\pmod {2}}\end{aligned}}}"></span></dd></dl> <p>Where <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 [b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e016d81a1b29bb79b275e446042b925b7dd6b6d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.291ex; height:2.843ex;" alt="{\displaystyle [b]}"></span> is the <a href="/wiki/Iverson_bracket" title="Iverson bracket">Iverson bracket</a>. </p><p>and working modulo <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 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}"></span> we can similarly prove that </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 {\begin{aligned}\left[{\begin{matrix}n\\m\end{matrix}}\right]&\equiv [x^{m}](x^{\lceil n/3\rceil }(x+1)^{\lceil (n-1)/3\rceil }(x+2)^{\lfloor n/3\rfloor }&&{\pmod {3}}\\&\equiv \sum _{k=0}^{m}{\binom {\lceil (n-1)/3\rceil }{k}}{\binom {\lfloor n/3\rfloor }{m-k-\lfloor n/3\rfloor }}2^{\lceil n/3\rceil +\lfloor n/3\rfloor -m+k}&&{\pmod {3}}\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> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>≡</mo> <mo stretchy="false">[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌈</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo fence="false" stretchy="false">⌉</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌈</mo> <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>3</mn> <mo fence="false" stretchy="false">⌉</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo fence="false" stretchy="false">⌋</mo> </mrow> </msup> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <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"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mo fence="false" stretchy="false">⌈</mo> <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>3</mn> <mo fence="false" stretchy="false">⌉</mo> </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"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo fence="false" stretchy="false">⌋</mo> </mrow> <mrow> <mi>m</mi> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo fence="false" stretchy="false">⌋</mo> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌈</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo fence="false" stretchy="false">⌉</mo> <mo>+</mo> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo fence="false" stretchy="false">⌋</mo> <mo>−</mo> <mi>m</mi> <mo>+</mo> <mi>k</mi> </mrow> </msup> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left[{\begin{matrix}n\\m\end{matrix}}\right]&\equiv [x^{m}](x^{\lceil n/3\rceil }(x+1)^{\lceil (n-1)/3\rceil }(x+2)^{\lfloor n/3\rfloor }&&{\pmod {3}}\\&\equiv \sum _{k=0}^{m}{\binom {\lceil (n-1)/3\rceil }{k}}{\binom {\lfloor n/3\rfloor }{m-k-\lfloor n/3\rfloor }}2^{\lceil n/3\rceil +\lfloor n/3\rfloor -m+k}&&{\pmod {3}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51dc5eb37a3814093035f02008dba120a24a7426" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.796ex; margin-bottom: -0.208ex; width:77.196ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\left[{\begin{matrix}n\\m\end{matrix}}\right]&\equiv [x^{m}](x^{\lceil n/3\rceil }(x+1)^{\lceil (n-1)/3\rceil }(x+2)^{\lfloor n/3\rfloor }&&{\pmod {3}}\\&\equiv \sum _{k=0}^{m}{\binom {\lceil (n-1)/3\rceil }{k}}{\binom {\lfloor n/3\rfloor }{m-k-\lfloor n/3\rfloor }}2^{\lceil n/3\rceil +\lfloor n/3\rfloor -m+k}&&{\pmod {3}}\end{aligned}}}"></span></dd></dl> <p>More generally, for fixed integers <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 h\geq 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>≥</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\geq 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22e2366ae4a94e16d6e8391d4f154eb843889bc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.6ex; height:2.343ex;" alt="{\displaystyle h\geq 3}"></span> if we define the ordered roots </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 \left(\omega _{h,i}\right)_{i=1}^{h-1}:=\left\{\omega _{j}:\sum _{i=0}^{h-1}{\binom {h-1}{i}}{\frac {h!}{(i+1)!}}(-\omega _{j})^{i}=0,\ 1\leq j<h\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow> <mo>(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>:=</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <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>h</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>h</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>i</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>h</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo><</mo> <mi>h</mi> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\omega _{h,i}\right)_{i=1}^{h-1}:=\left\{\omega _{j}:\sum _{i=0}^{h-1}{\binom {h-1}{i}}{\frac {h!}{(i+1)!}}(-\omega _{j})^{i}=0,\ 1\leq j<h\right\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6847203badcb543181b413e666def09def859992" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64.464ex; height:7.509ex;" alt="{\displaystyle \left(\omega _{h,i}\right)_{i=1}^{h-1}:=\left\{\omega _{j}:\sum _{i=0}^{h-1}{\binom {h-1}{i}}{\frac {h!}{(i+1)!}}(-\omega _{j})^{i}=0,\ 1\leq j<h\right\},}"></span></dd></dl> <p>then we may expand congruences for these Stirling numbers defined as the coefficients </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 \left[{\begin{matrix}n\\m\end{matrix}}\right]=[R^{m}]R(R+1)\cdots (R+n-1),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{matrix}n\\m\end{matrix}}\right]=[R^{m}]R(R+1)\cdots (R+n-1),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61992b653be1dffa456d368beef6a4c86e483f0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.374ex; height:6.176ex;" alt="{\displaystyle \left[{\begin{matrix}n\\m\end{matrix}}\right]=[R^{m}]R(R+1)\cdots (R+n-1),}"></span></dd></dl> <p>in the following form where the functions, <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 p_{h,i}^{[m]}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>,</mo> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{h,i}^{[m]}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d563739faa224f2a4ac4e60401e162f0c4c5fc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:7.053ex; height:4.009ex;" alt="{\displaystyle p_{h,i}^{[m]}(n)}"></span>, denote fixed polynomials of degree <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> in <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> for each <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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>, <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>, and <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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>: </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 \left[{\begin{matrix}n\\m\end{matrix}}\right]=\left(\sum _{i=0}^{h-1}p_{h,i}^{[m]}(n)\times \omega _{h,i}^{n}\right)[n>m]+[n=m]\qquad {\pmod {h}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>,</mo> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>,</mo> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">[</mo> <mi>n</mi> <mo>></mo> <mi>m</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo>=</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{matrix}n\\m\end{matrix}}\right]=\left(\sum _{i=0}^{h-1}p_{h,i}^{[m]}(n)\times \omega _{h,i}^{n}\right)[n>m]+[n=m]\qquad {\pmod {h}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2808e0e64c304265ae289188da7743e0423aa3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64.419ex; height:7.509ex;" alt="{\displaystyle \left[{\begin{matrix}n\\m\end{matrix}}\right]=\left(\sum _{i=0}^{h-1}p_{h,i}^{[m]}(n)\times \omega _{h,i}^{n}\right)[n>m]+[n=m]\qquad {\pmod {h}},}"></span></dd></dl> <p>Section 6.2 of the reference cited above provides more explicit expansions related to these congruences for the <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>-order <a href="/wiki/Harmonic_number" title="Harmonic number">harmonic numbers</a> and for the <a href="/wiki/Pochhammer_k-symbol" title="Pochhammer k-symbol">generalized factorial products</a>, <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 p_{n}(\alpha ,R):=R(R+\alpha )\cdots (R+(n-1)\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}(\alpha ,R):=R(R+\alpha )\cdots (R+(n-1)\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a434c9777e934d2bd6c5d683cbfbd3fad091a48f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:40.588ex; height:2.843ex;" alt="{\displaystyle p_{n}(\alpha ,R):=R(R+\alpha )\cdots (R+(n-1)\alpha )}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Generating_functions">Generating functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=12" title="Edit section: Generating functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A variety of identities may be derived by manipulating the <a href="/wiki/Generating_function" title="Generating function">generating function</a> (see <a href="/wiki/Stirling_number#As_change_of_basis_coefficients" title="Stirling number">change of basis</a>): </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 H(z,u)=(1+z)^{u}=\sum _{n=0}^{\infty }{u \choose n}z^{n}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\sum _{k=0}^{n}s(n,k)u^{k}=\sum _{k=0}^{\infty }u^{k}\sum _{n=k}^{\infty }{\frac {z^{n}}{n!}}s(n,k).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> <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 mathvariant="normal">∞</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>u</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <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 mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</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>n</mi> </mrow> </munderover> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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 mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(z,u)=(1+z)^{u}=\sum _{n=0}^{\infty }{u \choose n}z^{n}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\sum _{k=0}^{n}s(n,k)u^{k}=\sum _{k=0}^{\infty }u^{k}\sum _{n=k}^{\infty }{\frac {z^{n}}{n!}}s(n,k).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f0a3dd483098b62ffda9384a86c5627ad994c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:79.253ex; height:7.009ex;" alt="{\displaystyle H(z,u)=(1+z)^{u}=\sum _{n=0}^{\infty }{u \choose n}z^{n}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\sum _{k=0}^{n}s(n,k)u^{k}=\sum _{k=0}^{\infty }u^{k}\sum _{n=k}^{\infty }{\frac {z^{n}}{n!}}s(n,k).}"></span></dd></dl> <p>Using the equality </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+z)^{u}=e^{u\log(1+z)}=\sum _{k=0}^{\infty }(\log(1+z))^{k}{\frac {u^{k}}{k!}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </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 mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>u</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 (1+z)^{u}=e^{u\log(1+z)}=\sum _{k=0}^{\infty }(\log(1+z))^{k}{\frac {u^{k}}{k!}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a67ffc5d26d8299bb1ea8cc55d23dc90d30ec170" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.19ex; height:7.009ex;" alt="{\displaystyle (1+z)^{u}=e^{u\log(1+z)}=\sum _{k=0}^{\infty }(\log(1+z))^{k}{\frac {u^{k}}{k!}},}"></span></dd></dl> <p>it follows that </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 \sum _{n=k}^{\infty }s(n,k){\frac {z^{n}}{n!}}={\frac {(\log(1+z))^{k}}{k!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=k}^{\infty }s(n,k){\frac {z^{n}}{n!}}={\frac {(\log(1+z))^{k}}{k!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a195d2677f0524774bc3d999547d6c888a5c1e52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.131ex; height:7.009ex;" alt="{\displaystyle \sum _{n=k}^{\infty }s(n,k){\frac {z^{n}}{n!}}={\frac {(\log(1+z))^{k}}{k!}}}"></span></dd></dl> <p>and </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 \sum _{n=k}^{\infty }\left[{n \atop k}\right]{\frac {z^{n}}{n!}}={\frac {(-\log(1-z))^{k}}{k!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=k}^{\infty }\left[{n \atop k}\right]{\frac {z^{n}}{n!}}={\frac {(-\log(1-z))^{k}}{k!}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77f15a752f1238a639c3b9f3b85809cb35776ad8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.968ex; height:7.009ex;" alt="{\displaystyle \sum _{n=k}^{\infty }\left[{n \atop k}\right]{\frac {z^{n}}{n!}}={\frac {(-\log(1-z))^{k}}{k!}}.}"></span><sup id="cite_ref-:0_1-2" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></dd></dl> <p>This identity is valid for <a href="/wiki/Formal_power_series" title="Formal power series">formal power series</a>, and the sum <a href="/wiki/Convergent_series" title="Convergent series">converges</a> in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> for |<i>z</i>| < 1. </p><p>Other identities arise by exchanging the order of summation, taking derivatives, making substitutions for <i>z</i> or <i>u</i>, etc. For example, we may derive:<sup id="cite_ref-blagouch1_14-0" class="reference"><a href="#cite_note-blagouch1-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </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 {\frac {\log ^{m}(1+z)}{1+z}}=m!\sum _{k=0}^{\infty }{\frac {s(k+1,m+1)\,z^{k}}{k!}},\qquad m=1,2,3,\ldots \quad |z|<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>z</mi> </mrow> </mfrac> </mrow> <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 mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>m</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\log ^{m}(1+z)}{1+z}}=m!\sum _{k=0}^{\infty }{\frac {s(k+1,m+1)\,z^{k}}{k!}},\qquad m=1,2,3,\ldots \quad |z|<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5b85dba9571c8caba00179793c5992cee580d8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:70.374ex; height:7.176ex;" alt="{\displaystyle {\frac {\log ^{m}(1+z)}{1+z}}=m!\sum _{k=0}^{\infty }{\frac {s(k+1,m+1)\,z^{k}}{k!}},\qquad m=1,2,3,\ldots \quad |z|<1}"></span></dd></dl> <p>or </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 \sum _{n=i}^{\infty }{\frac {\left[{n \atop i}\right]}{n\,(n!)}}=\zeta (i+1),\qquad i=1,2,3,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>i</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow> <mi>n</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <mi>i</mi> <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 \sum _{n=i}^{\infty }{\frac {\left[{n \atop i}\right]}{n\,(n!)}}=\zeta (i+1),\qquad i=1,2,3,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f73d41945e57eeceecf450c3ca49cba2d4e8805e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.911ex; height:7.343ex;" alt="{\displaystyle \sum _{n=i}^{\infty }{\frac {\left[{n \atop i}\right]}{n\,(n!)}}=\zeta (i+1),\qquad i=1,2,3,\ldots }"></span></dd></dl> <p>and </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 \sum _{n=i}^{\infty }{\frac {\left[{n \atop i}\right]}{n\,(v)_{n}}}=\zeta (i+1,v),\qquad i=1,2,3,\ldots \quad \Re (v)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>i</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow> <mi>n</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>v</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mspace width="1em" /> <mi mathvariant="normal">ℜ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=i}^{\infty }{\frac {\left[{n \atop i}\right]}{n\,(v)_{n}}}=\zeta (i+1,v),\qquad i=1,2,3,\ldots \quad \Re (v)>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a5811fa18741e5b10e8b4cc2ad1836298600595" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:54.209ex; height:7.343ex;" alt="{\displaystyle \sum _{n=i}^{\infty }{\frac {\left[{n \atop i}\right]}{n\,(v)_{n}}}=\zeta (i+1,v),\qquad i=1,2,3,\ldots \quad \Re (v)>0}"></span></dd></dl> <p>where <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 \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></span><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)}"></span> and <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 \zeta (k,v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (k,v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7101eab920168c30b14abf77ab75b0dcd3e263b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.277ex; height:2.843ex;" alt="{\displaystyle \zeta (k,v)}"></span> are the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> and the <a href="/wiki/Hurwitz_zeta_function" title="Hurwitz zeta function">Hurwitz zeta function</a> respectively, and even evaluate this integral </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 \int _{0}^{1}{\frac {\log ^{z}(1-x)}{x^{k}}}\,dx={\frac {(-1)^{z}\Gamma (z+1)}{(k-1)!}}\sum _{r=1}^{k-1}s(k-1,r)\sum _{m=0}^{r}{\binom {r}{m}}(k-2)^{r-m}\zeta (z+1-m),\qquad \Re (z)>k-1,\quad k=3,4,5,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <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>z</mi> </mrow> </msup> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>s</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</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>r</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>−</mo> <mi>m</mi> </mrow> </msup> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <mi mathvariant="normal">ℜ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em" /> <mi>k</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}{\frac {\log ^{z}(1-x)}{x^{k}}}\,dx={\frac {(-1)^{z}\Gamma (z+1)}{(k-1)!}}\sum _{r=1}^{k-1}s(k-1,r)\sum _{m=0}^{r}{\binom {r}{m}}(k-2)^{r-m}\zeta (z+1-m),\qquad \Re (z)>k-1,\quad k=3,4,5,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34f4915133e60a35029aa3ae6adb6fba14f24859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:120.35ex; height:7.343ex;" alt="{\displaystyle \int _{0}^{1}{\frac {\log ^{z}(1-x)}{x^{k}}}\,dx={\frac {(-1)^{z}\Gamma (z+1)}{(k-1)!}}\sum _{r=1}^{k-1}s(k-1,r)\sum _{m=0}^{r}{\binom {r}{m}}(k-2)^{r-m}\zeta (z+1-m),\qquad \Re (z)>k-1,\quad k=3,4,5,\ldots }"></span></dd></dl> <p>where <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 (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11ca17f880240539116aac7e6326909299e2a080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.35ex; height:2.843ex;" alt="{\displaystyle \Gamma (z)}"></span> is the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>. There also exist more complicated expressions for the zeta-functions involving the Stirling numbers. One, for example, has </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 \zeta (s,v)={\frac {k!}{(s-k)_{k}}}\sum _{n=0}^{\infty }{\frac {1}{(n+k)!}}\left[{n+k \atop n}\right]\sum _{l=0}^{n+k-1}\!(-1)^{l}{\binom {n+k-1}{l}}(l+v)^{k-s},\quad k=1,2,3,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>k</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </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> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mspace width="negativethinmathspace" /> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</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>n</mi> <mo>+</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>l</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mi>v</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>k</mi> <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 \zeta (s,v)={\frac {k!}{(s-k)_{k}}}\sum _{n=0}^{\infty }{\frac {1}{(n+k)!}}\left[{n+k \atop n}\right]\sum _{l=0}^{n+k-1}\!(-1)^{l}{\binom {n+k-1}{l}}(l+v)^{k-s},\quad k=1,2,3,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/836cd9b85d9e531a42385942aa050ea2ce7fa183" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:90.027ex; height:7.509ex;" alt="{\displaystyle \zeta (s,v)={\frac {k!}{(s-k)_{k}}}\sum _{n=0}^{\infty }{\frac {1}{(n+k)!}}\left[{n+k \atop n}\right]\sum _{l=0}^{n+k-1}\!(-1)^{l}{\binom {n+k-1}{l}}(l+v)^{k-s},\quad k=1,2,3,\ldots }"></span></dd></dl> <p>This series generalizes <a href="/wiki/Helmut_Hasse" title="Helmut Hasse">Hasse</a>'s series for the <a href="/wiki/Hurwitz_zeta-function" class="mw-redirect" title="Hurwitz zeta-function">Hurwitz zeta-function</a> (we obtain Hasse's series by setting <i>k</i>=1).<sup id="cite_ref-blag2018_15-0" class="reference"><a href="#cite_note-blag2018-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Asymptotics">Asymptotics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=13" title="Edit section: Asymptotics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The next estimate given in terms of the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler gamma constant</a> applies:<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </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 \left[{\begin{matrix}n+1\\k+1\end{matrix}}\right]{\underset {n\to \infty }{\sim }}{\frac {n!}{k!}}\left(\gamma +\ln n\right)^{k},\ {\text{ uniformly for }}k=o(\ln n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∼</mo> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>!</mo> </mrow> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>γ</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> uniformly for </mtext> </mrow> <mi>k</mi> <mo>=</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{matrix}n+1\\k+1\end{matrix}}\right]{\underset {n\to \infty }{\sim }}{\frac {n!}{k!}}\left(\gamma +\ln n\right)^{k},\ {\text{ uniformly for }}k=o(\ln n).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9128052e6be63058a3751b14e6356bfa504bcdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:54.448ex; height:6.176ex;" alt="{\displaystyle \left[{\begin{matrix}n+1\\k+1\end{matrix}}\right]{\underset {n\to \infty }{\sim }}{\frac {n!}{k!}}\left(\gamma +\ln n\right)^{k},\ {\text{ uniformly for }}k=o(\ln n).}"></span></dd></dl> <p>For fixed <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> we have the following estimate : </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 \left[{\begin{matrix}n+k\\k\end{matrix}}\right]{\underset {k\to \infty }{\sim }}{\frac {k^{2n}}{2^{n}n!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo>∼</mo> <mrow> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{matrix}n+k\\k\end{matrix}}\right]{\underset {k\to \infty }{\sim }}{\frac {k^{2n}}{2^{n}n!}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae039f7a357016ddda364152641bb5a32f48e844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.088ex; height:6.343ex;" alt="{\displaystyle \left[{\begin{matrix}n+k\\k\end{matrix}}\right]{\underset {k\to \infty }{\sim }}{\frac {k^{2n}}{2^{n}n!}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Explicit_formula">Explicit formula</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=14" title="Edit section: Explicit formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is well-known that we don't know any one-sum formula for Stirling numbers of the first kind. A two-sum formula can be obtained using one of the <a href="/wiki/Stirling_number#Symmetric_formulae" title="Stirling number">symmetric formulae for Stirling numbers</a> in conjunction with the explicit formula for <a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling numbers of the second kind</a>. </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 \left[{n \atop k}\right]=\sum _{j=n}^{2n-k}{\binom {j-1}{k-1}}{\binom {2n-k}{j}}\sum _{m=0}^{j-n}{\frac {(-1)^{m+n-k}m^{j-k}}{m!(j-n-m)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <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"> <mrow> <mi>j</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="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </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> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> <mi>j</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>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mi>n</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>m</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <mo>!</mo> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{n \atop k}\right]=\sum _{j=n}^{2n-k}{\binom {j-1}{k-1}}{\binom {2n-k}{j}}\sum _{m=0}^{j-n}{\frac {(-1)^{m+n-k}m^{j-k}}{m!(j-n-m)!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf870a1fe3594bf66af35f4d27a24e6fbbb1f492" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:51.334ex; height:7.676ex;" alt="{\displaystyle \left[{n \atop k}\right]=\sum _{j=n}^{2n-k}{\binom {j-1}{k-1}}{\binom {2n-k}{j}}\sum _{m=0}^{j-n}{\frac {(-1)^{m+n-k}m^{j-k}}{m!(j-n-m)!}}}"></span></dd></dl> <p>As discussed earlier, by <a href="/wiki/Vieta%27s_formulas" title="Vieta's formulas">Vieta's formulas</a>, one get<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]=\sum _{0\leq i_{1}<\ldots <i_{n-k}<n}i_{1}i_{2}\cdots i_{n-k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>≤</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>…</mo> <mo><</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo><</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]=\sum _{0\leq i_{1}<\ldots <i_{n-k}<n}i_{1}i_{2}\cdots i_{n-k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a801dbd28ee4a947b9efbd5d40fdadfc7a51eba6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:33.613ex; height:7.176ex;" alt="{\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]=\sum _{0\leq i_{1}<\ldots <i_{n-k}<n}i_{1}i_{2}\cdots i_{n-k}.}"></span>The Stirling number <i>s(n,n-p)</i> can be found from the formula<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </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 {\begin{aligned}s(n,n-p)&={\frac {1}{(n-p-1)!}}\sum _{0\leq k_{1},\ldots ,k_{p}:\sum _{1}^{p}mk_{m}=p}(-1)^{K}{\frac {(n+K-1)!}{k_{1}!k_{2}!\cdots k_{p}!~2!^{k_{1}}3!^{k_{2}}\cdots (p+1)!^{k_{p}}}},\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>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>≤</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>:</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <mi>m</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mi>p</mi> </mrow> </munder> <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> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>K</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>!</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>!</mo> <mo>⋯</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>!</mo> <mtext> </mtext> <mn>2</mn> <msup> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mn>3</mn> <msup> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mo>!</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s(n,n-p)&={\frac {1}{(n-p-1)!}}\sum _{0\leq k_{1},\ldots ,k_{p}:\sum _{1}^{p}mk_{m}=p}(-1)^{K}{\frac {(n+K-1)!}{k_{1}!k_{2}!\cdots k_{p}!~2!^{k_{1}}3!^{k_{2}}\cdots (p+1)!^{k_{p}}}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/068759dd90e1cf0e061c587d3a559b018a3a5305" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:86.431ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}s(n,n-p)&={\frac {1}{(n-p-1)!}}\sum _{0\leq k_{1},\ldots ,k_{p}:\sum _{1}^{p}mk_{m}=p}(-1)^{K}{\frac {(n+K-1)!}{k_{1}!k_{2}!\cdots k_{p}!~2!^{k_{1}}3!^{k_{2}}\cdots (p+1)!^{k_{p}}}},\end{aligned}}}"></span></dd></dl> <p>where <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 K=k_{1}+\cdots +k_{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=k_{1}+\cdots +k_{p}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc3272c7f356db00a93edcfd8f35d21a06a23d61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.751ex; height:2.843ex;" alt="{\displaystyle K=k_{1}+\cdots +k_{p}.}"></span> The sum is a sum over all <a href="/wiki/Partition_(number_theory)" class="mw-redirect" title="Partition (number theory)">partitions</a> of <i>p</i>. </p><p>Another exact nested sum expansion for these Stirling numbers is computed by <a href="/wiki/Elementary_symmetric_polynomials" class="mw-redirect" title="Elementary symmetric polynomials">elementary symmetric polynomials</a> corresponding to the coefficients in <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> of a product of the form <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+c_{1}x)\cdots (1+c_{n-1}x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+c_{1}x)\cdots (1+c_{n-1}x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d108175effc3e3a3e3c097403f7e5932a3fc50c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.168ex; height:2.843ex;" alt="{\displaystyle (1+c_{1}x)\cdots (1+c_{n-1}x)}"></span>. In particular, we see that </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 {\begin{aligned}\left[{n \atop k+1}\right]&=[x^{k}](x+1)(x+2)\cdots (x+n-1)=(n-1)!\cdot [x^{k}](x+1)\left({\frac {x}{2}}+1\right)\cdots \left({\frac {x}{n-1}}+1\right)\\&=\sum _{1\leq i_{1}<\cdots <i_{k}<n}{\frac {(n-1)!}{i_{1}\cdots i_{k}}}.\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> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>⋅</mo> <mo stretchy="false">[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo><</mo> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left[{n \atop k+1}\right]&=[x^{k}](x+1)(x+2)\cdots (x+n-1)=(n-1)!\cdot [x^{k}](x+1)\left({\frac {x}{2}}+1\right)\cdots \left({\frac {x}{n-1}}+1\right)\\&=\sum _{1\leq i_{1}<\cdots <i_{k}<n}{\frac {(n-1)!}{i_{1}\cdots i_{k}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12edcc3b990c2590254ecbf2340492bc86fe4551" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:94.918ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}\left[{n \atop k+1}\right]&=[x^{k}](x+1)(x+2)\cdots (x+n-1)=(n-1)!\cdot [x^{k}](x+1)\left({\frac {x}{2}}+1\right)\cdots \left({\frac {x}{n-1}}+1\right)\\&=\sum _{1\leq i_{1}<\cdots <i_{k}<n}{\frac {(n-1)!}{i_{1}\cdots i_{k}}}.\end{aligned}}}"></span></dd></dl> <p><a href="/wiki/Newton%27s_identities" title="Newton's identities">Newton's identities</a> combined with the above expansions may be used to give an alternate proof of the weighted expansions involving the generalized <a href="/wiki/Harmonic_number" title="Harmonic number">harmonic numbers</a> already <a href="#Expansions_for_fixed_k">noted above</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Relations_to_natural_logarithm_function">Relations to natural logarithm function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=15" title="Edit section: Relations to natural logarithm function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>n</i>th <a href="/wiki/Derivative" title="Derivative">derivative</a> of the <i>μ</i>th power of the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> involves the signed Stirling numbers of the first kind: </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 {\operatorname {d} ^{n}\!(\ln x)^{\mu } \over \operatorname {d} \!x^{n}}=x^{-n}\sum _{k=1}^{n}\mu ^{\underline {k}}s(n,n-k+1)(\ln x)^{\mu -k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="negativethinmathspace" /> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mspace width="negativethinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msup> <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>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>k</mi> <mo>_</mo> </munder> </mrow> </msup> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\operatorname {d} ^{n}\!(\ln x)^{\mu } \over \operatorname {d} \!x^{n}}=x^{-n}\sum _{k=1}^{n}\mu ^{\underline {k}}s(n,n-k+1)(\ln x)^{\mu -k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b125c8985dc20758e0c449e3e57b8ac78b4698" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.444ex; height:6.843ex;" alt="{\displaystyle {\operatorname {d} ^{n}\!(\ln x)^{\mu } \over \operatorname {d} \!x^{n}}=x^{-n}\sum _{k=1}^{n}\mu ^{\underline {k}}s(n,n-k+1)(\ln x)^{\mu -k},}"></span> </p><p>where <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 \mu ^{\underline {i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>i</mi> <mo>_</mo> </munder> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ^{\underline {i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0036f93f785c7c40775a65444d57f7b68347ff9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:3.176ex;" alt="{\displaystyle \mu ^{\underline {i}}}"></span> is the <a href="/wiki/Falling_factorial" class="mw-redirect" title="Falling factorial">falling factorial</a>, and <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 s(n,n-k+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(n,n-k+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab7a46263212b79551ecc822b8c1c7fe4f14daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.778ex; height:2.843ex;" alt="{\displaystyle s(n,n-k+1)}"></span> is the signed Stirling number. </p><p>It can be proved by using <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Other_formulas">Other formulas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=16" title="Edit section: Other formulas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Stirling numbers of the first kind appear in the formula for <a href="/wiki/Gregory_coefficients" title="Gregory coefficients">Gregory coefficients</a> and in a finite sum identity involving <a href="/wiki/Bell_number" title="Bell number">Bell numbers</a><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </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 n!G_{n}=\sum _{l=0}^{n}{\frac {s(n,l)}{l+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>l</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>l</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!G_{n}=\sum _{l=0}^{n}{\frac {s(n,l)}{l+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/752067e63fc193a6b83a45ce679a209c1c5a3658" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.785ex; height:7.009ex;" alt="{\displaystyle n!G_{n}=\sum _{l=0}^{n}{\frac {s(n,l)}{l+1}}}"></span> </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 \sum _{j=0}^{n}{\binom {n}{j}}B_{j}k^{n-j}=\sum _{i=0}^{k}\left[{k \atop i}\right]B_{n+i}(-1)^{k-i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</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>j</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>j</mi> </mrow> </msub> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>j</mi> </mrow> </msup> <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>k</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>k</mi> <mi>i</mi> </mfrac> </mrow> <mo>]</mo> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j=0}^{n}{\binom {n}{j}}B_{j}k^{n-j}=\sum _{i=0}^{k}\left[{k \atop i}\right]B_{n+i}(-1)^{k-i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09efee128e30bd53f1153a352f331a7e582b73ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:39.612ex; height:7.676ex;" alt="{\displaystyle \sum _{j=0}^{n}{\binom {n}{j}}B_{j}k^{n-j}=\sum _{i=0}^{k}\left[{k \atop i}\right]B_{n+i}(-1)^{k-i}}"></span> </p><p>Infinite series involving the finite sums with the Stirling numbers often lead to the special functions. For example<sup id="cite_ref-blagouch1_14-1" class="reference"><a href="#cite_note-blagouch1-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </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 \Gamma (z)=\left(z-{\frac {1}{2}}\right)\!\ln z-z+{\frac {1}{2}}\ln 2\pi +{\frac {1}{\pi }}\sum _{n=1}^{\infty }{\frac {1}{n\cdot n!}}\!\sum _{l=0}^{\lfloor n/2\rfloor }\!{\frac {(-1)^{l}(2l)!}{(2\pi z)^{2l+1}}}\left[{n \atop 2l+1}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="negativethinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> <mo>−</mo> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> <mi>π</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </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> <mn>1</mn> <mrow> <mi>n</mi> <mo>⋅</mo> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mspace width="negativethinmathspace" /> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌋</mo> </mrow> </munderover> <mspace width="negativethinmathspace" /> <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>l</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>l</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>l</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mn>2</mn> <mi>l</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \Gamma (z)=\left(z-{\frac {1}{2}}\right)\!\ln z-z+{\frac {1}{2}}\ln 2\pi +{\frac {1}{\pi }}\sum _{n=1}^{\infty }{\frac {1}{n\cdot n!}}\!\sum _{l=0}^{\lfloor n/2\rfloor }\!{\frac {(-1)^{l}(2l)!}{(2\pi z)^{2l+1}}}\left[{n \atop 2l+1}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5369003b89279cc3a83866243e9c34945eec5f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:75.601ex; height:7.843ex;" alt="{\displaystyle \ln \Gamma (z)=\left(z-{\frac {1}{2}}\right)\!\ln z-z+{\frac {1}{2}}\ln 2\pi +{\frac {1}{\pi }}\sum _{n=1}^{\infty }{\frac {1}{n\cdot n!}}\!\sum _{l=0}^{\lfloor n/2\rfloor }\!{\frac {(-1)^{l}(2l)!}{(2\pi z)^{2l+1}}}\left[{n \atop 2l+1}\right]}"></span> </p><p>and </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 \Psi (z)=\ln z-{\frac {1}{2z}}-{\frac {1}{\pi z}}\sum _{n=1}^{\infty }{\frac {1}{n\cdot n!}}\!\sum _{l=0}^{\lfloor n/2\rfloor }\!{\frac {(-1)^{l}(2l+1)!}{(2\pi z)^{2l+1}}}\left[{n \atop 2l+1}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>π</mi> <mi>z</mi> </mrow> </mfrac> </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> <mn>1</mn> <mrow> <mi>n</mi> <mo>⋅</mo> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mspace width="negativethinmathspace" /> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌋</mo> </mrow> </munderover> <mspace width="negativethinmathspace" /> <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>l</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>l</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mn>2</mn> <mi>l</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (z)=\ln z-{\frac {1}{2z}}-{\frac {1}{\pi z}}\sum _{n=1}^{\infty }{\frac {1}{n\cdot n!}}\!\sum _{l=0}^{\lfloor n/2\rfloor }\!{\frac {(-1)^{l}(2l+1)!}{(2\pi z)^{2l+1}}}\left[{n \atop 2l+1}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baffbb1865f637881e0593c4685a26149fb834e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:61.325ex; height:7.843ex;" alt="{\displaystyle \Psi (z)=\ln z-{\frac {1}{2z}}-{\frac {1}{\pi z}}\sum _{n=1}^{\infty }{\frac {1}{n\cdot n!}}\!\sum _{l=0}^{\lfloor n/2\rfloor }\!{\frac {(-1)^{l}(2l+1)!}{(2\pi z)^{2l+1}}}\left[{n \atop 2l+1}\right]}"></span> </p><p>or even </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 \gamma _{m}={\frac {1}{2}}\delta _{m,0}+{\frac {(-1)^{m}m!}{\pi }}\!\sum _{n=1}^{\infty }{\frac {1}{n\cdot n!}}\!\sum _{k=0}^{\lfloor n/2\rfloor }{\frac {(-1)^{k}}{(2\pi )^{2k+1}}}\left[{2k+2 \atop m+1}\right]\left[{n \atop 2k+1}\right]\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mn>0</mn> </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>m</mi> </mrow> </msup> <mi>m</mi> <mo>!</mo> </mrow> <mi>π</mi> </mfrac> </mrow> <mspace width="negativethinmathspace" /> <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> <mn>1</mn> <mrow> <mi>n</mi> <mo>⋅</mo> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mspace width="negativethinmathspace" /> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌋</mo> </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> <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> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{m}={\frac {1}{2}}\delta _{m,0}+{\frac {(-1)^{m}m!}{\pi }}\!\sum _{n=1}^{\infty }{\frac {1}{n\cdot n!}}\!\sum _{k=0}^{\lfloor n/2\rfloor }{\frac {(-1)^{k}}{(2\pi )^{2k+1}}}\left[{2k+2 \atop m+1}\right]\left[{n \atop 2k+1}\right]\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f341b955c5e9736a48c3192879e5478ff16212d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:68.043ex; height:7.843ex;" alt="{\displaystyle \gamma _{m}={\frac {1}{2}}\delta _{m,0}+{\frac {(-1)^{m}m!}{\pi }}\!\sum _{n=1}^{\infty }{\frac {1}{n\cdot n!}}\!\sum _{k=0}^{\lfloor n/2\rfloor }{\frac {(-1)^{k}}{(2\pi )^{2k+1}}}\left[{2k+2 \atop m+1}\right]\left[{n \atop 2k+1}\right]\,}"></span> </p><p>where <i>γ</i><sub><i>m</i></sub> are the <a href="/wiki/Stieltjes_constants" title="Stieltjes constants">Stieltjes constants</a> and <i>δ</i><sub><i>m</i>,0</sub> represents the <a href="/wiki/Kronecker_delta_function" class="mw-redirect" title="Kronecker delta function">Kronecker delta function</a>. </p><p>Notice that this last identity immediately implies relations between the <a href="/wiki/Polylogarithm" title="Polylogarithm">polylogarithm</a> functions, the Stirling number exponential <a href="/wiki/Generating_function" title="Generating function">generating functions</a> given above, and the Stirling-number-based power series for the generalized <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/NielsenGeneralizedPolylogarithm.html">Nielsen polylogarithm</a> functions. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=17" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are many notions of <i>generalized Stirling numbers</i> that may be defined (depending on application) in a number of differing combinatorial contexts. In so much as the Stirling numbers of the first kind correspond to the coefficients of the distinct polynomial expansions of the <a href="/wiki/Factorial" title="Factorial">single factorial function</a>, <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)(n-2)\cdots 2\cdot 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!=n(n-1)(n-2)\cdots 2\cdot 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcbf4e885e0893c0142027ecd1c451986423c511" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.45ex; height:2.843ex;" alt="{\displaystyle n!=n(n-1)(n-2)\cdots 2\cdot 1}"></span>, we may extend this notion to define triangular recurrence relations for more general classes of products. </p><p>In particular, for any fixed arithmetic function <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 f:\mathbb {N} \rightarrow \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {N} \rightarrow \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6259a2fa90a116947f51109253abd3ebb517b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {N} \rightarrow \mathbb {C} }"></span> and symbolic parameters <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 x,t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef10a0c06f8a5238d439b9a7bde431605db5190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.203ex; height:2.343ex;" alt="{\displaystyle x,t}"></span>, related generalized factorial products of the form </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 (x)_{n,f,t}:=\prod _{k=1}^{n-1}\left(x+{\frac {f(k)}{t^{k}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>f</mi> <mo>,</mo> <mi>t</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> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x)_{n,f,t}:=\prod _{k=1}^{n-1}\left(x+{\frac {f(k)}{t^{k}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/910c7a27fa90f7396bc1246a1001b7619387dcd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.716ex; height:7.343ex;" alt="{\displaystyle (x)_{n,f,t}:=\prod _{k=1}^{n-1}\left(x+{\frac {f(k)}{t^{k}}}\right)}"></span></dd></dl> <p>may be studied from the point of view of the classes of generalized Stirling numbers of the first kind defined by the following coefficients of the powers of <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> in the expansions of <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 (x)_{n,f,t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>f</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x)_{n,f,t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b2a3d4edb11ace0910cc3926c45effce5bcadee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.77ex; height:3.009ex;" alt="{\displaystyle (x)_{n,f,t}}"></span> and then by the next corresponding triangular recurrence relation: </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 {\begin{aligned}\left[{\begin{matrix}n\\k\end{matrix}}\right]_{f,t}&=[x^{k-1}](x)_{n,f,t}\\&=f(n-1)t^{1-n}\left[{\begin{matrix}n-1\\k\end{matrix}}\right]_{f,t}+\left[{\begin{matrix}n-1\\k-1\end{matrix}}\right]_{f,t}+\delta _{n,0}\delta _{k,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> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>f</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>n</mi> </mrow> </msup> <msub> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left[{\begin{matrix}n\\k\end{matrix}}\right]_{f,t}&=[x^{k-1}](x)_{n,f,t}\\&=f(n-1)t^{1-n}\left[{\begin{matrix}n-1\\k\end{matrix}}\right]_{f,t}+\left[{\begin{matrix}n-1\\k-1\end{matrix}}\right]_{f,t}+\delta _{n,0}\delta _{k,0}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ada6ce80b7273d77af8ff6c78064d9683827c4db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:58.124ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\left[{\begin{matrix}n\\k\end{matrix}}\right]_{f,t}&=[x^{k-1}](x)_{n,f,t}\\&=f(n-1)t^{1-n}\left[{\begin{matrix}n-1\\k\end{matrix}}\right]_{f,t}+\left[{\begin{matrix}n-1\\k-1\end{matrix}}\right]_{f,t}+\delta _{n,0}\delta _{k,0}.\end{aligned}}}"></span></dd></dl> <p>These coefficients satisfy a number of analogous properties to those for the Stirling numbers of the first kind as well as recurrence relations and functional equations related to the <i>f-harmonic numbers</i>, <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 F_{n}^{(r)}(t):=\sum _{k\leq n}t^{k}/f(k)^{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}^{(r)}(t):=\sum _{k\leq n}t^{k}/f(k)^{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/929d0abd656bae697221e3d441d3c15ee0e05085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.568ex; height:6.009ex;" alt="{\displaystyle F_{n}^{(r)}(t):=\sum _{k\leq n}t^{k}/f(k)^{r}}"></span>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>One special case of these bracketed coefficients corresponding to <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 t\equiv 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>≡</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\equiv 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/575ac2758c41a01a283e9317736e13d9f9aa76c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t\equiv 1}"></span> allows us to expand the multiple factorial, or <a href="/wiki/Double_factorial#Generalizations" title="Double factorial">multifactorial</a> functions as polynomials in <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>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Stirling_numbers" class="mw-redirect" title="Stirling numbers">Stirling numbers</a> of both kinds, the <a href="/wiki/Binomial_coefficients" class="mw-redirect" title="Binomial coefficients">binomial coefficients</a>, and the first and second-order <a href="/wiki/Eulerian_numbers" class="mw-redirect" title="Eulerian numbers">Eulerian numbers</a> are all defined by special cases of a triangular <i>super-recurrence</i> of the form </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 \left|{\begin{matrix}n\\k\end{matrix}}\right|=(\alpha n+\beta k+\gamma )\left|{\begin{matrix}n-1\\k\end{matrix}}\right|+(\alpha ^{\prime }n+\beta ^{\prime }k+\gamma ^{\prime })\left|{\begin{matrix}n-1\\k-1\end{matrix}}\right|+\delta _{n,0}\delta _{k,0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>n</mi> <mo>+</mo> <mi>β</mi> <mi>k</mi> <mo>+</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mi>n</mi> <mo>+</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mi>k</mi> <mo>+</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\begin{matrix}n\\k\end{matrix}}\right|=(\alpha n+\beta k+\gamma )\left|{\begin{matrix}n-1\\k\end{matrix}}\right|+(\alpha ^{\prime }n+\beta ^{\prime }k+\gamma ^{\prime })\left|{\begin{matrix}n-1\\k-1\end{matrix}}\right|+\delta _{n,0}\delta _{k,0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f77ddf598225dc76807466f310eb371004b97c94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:65.889ex; height:6.176ex;" alt="{\displaystyle \left|{\begin{matrix}n\\k\end{matrix}}\right|=(\alpha n+\beta k+\gamma )\left|{\begin{matrix}n-1\\k\end{matrix}}\right|+(\alpha ^{\prime }n+\beta ^{\prime }k+\gamma ^{\prime })\left|{\begin{matrix}n-1\\k-1\end{matrix}}\right|+\delta _{n,0}\delta _{k,0},}"></span></dd></dl> <p>for integers <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\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,k\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef7a901b2948d91b9b1df337d990cb7e878532b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.901ex; height:2.509ex;" alt="{\displaystyle n,k\geq 0}"></span> and where <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 \left|{\begin{matrix}n\\k\end{matrix}}\right|\equiv 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>≡</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\begin{matrix}n\\k\end{matrix}}\right|\equiv 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/385cca151dd658466efc76ca4943b91bc18a38d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.701ex; height:6.176ex;" alt="{\displaystyle \left|{\begin{matrix}n\\k\end{matrix}}\right|\equiv 0}"></span> whenever <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<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad9f68fb8426c4411972241aac6c359e7812eaba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n<0}"></span> or <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 k<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d59e54fad8568e90715f2b10521d3e39bc45fca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k<0}"></span>. In this sense, the form of the Stirling numbers of the first kind may also be generalized by this parameterized super-recurrence for fixed scalars <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 \alpha ,\beta ,\gamma ,\alpha ^{\prime },\beta ^{\prime },\gamma ^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> <mo>,</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta ,\gamma ,\alpha ^{\prime },\beta ^{\prime },\gamma ^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7ec5a2e8dcdc83f430f78c6ccf360f01a22f472" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.41ex; height:3.009ex;" alt="{\displaystyle \alpha ,\beta ,\gamma ,\alpha ^{\prime },\beta ^{\prime },\gamma ^{\prime }}"></span> (not all zero). </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Stirling_numbers" class="mw-redirect" title="Stirling numbers">Stirling numbers</a></li> <li><a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling numbers of the second kind</a></li> <li><a href="/wiki/Stirling_polynomials" title="Stirling polynomials">Stirling polynomials</a></li> <li><a href="/wiki/Random_permutation_statistics" title="Random permutation statistics">Random permutation statistics</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stirling_numbers_of_the_first_kind&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><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="CITEREFWilf1990" class="citation book cs1">Wilf, Herbert S. (1990). <i>Generatingfunctionology</i>. San Diego, CA, USA: Academic Press. p. 73. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-148324857-8" title="Special:BookSources/978-148324857-8"><bdi>978-148324857-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Generatingfunctionology&rft.place=San+Diego%2C+CA%2C+USA&rft.pages=73&rft.pub=Academic+Press&rft.date=1990&rft.isbn=978-148324857-8&rft.aulast=Wilf&rft.aufirst=Herbert+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: date and year (<a href="/wiki/Category:CS1_maint:_date_and_year" title="Category:CS1 maint: date and year">link</a>)</span></span> </li> <li id="cite_note-:1-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1992" class="citation journal cs1">Knuth, Donald E. (1992). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2325085">"Two Notes on Notation"</a>. <i>American Mathematical Monthly</i>. <b>99</b> (5): 403–422 – via JSTOR.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Two+Notes+on+Notation&rft.volume=99&rft.issue=5&rft.pages=403-422&rft.date=1992&rft.aulast=Knuth&rft.aufirst=Donald+E.&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2325085&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRényi1962" class="citation journal cs1 cs1-prop-long-vol">Rényi, Alfred (1962). <a rel="nofollow" class="external text" href="http://www.numdam.org/item/ASCFM_1962__8_2_7_0/">"Théorie des éléments saillants d'une suite d'observations"</a>. <i>Annales scientifiques de l'Université de Clermont. Mathématiques</i>. Tome 8 (2): 7–13.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+scientifiques+de+l%27Universit%C3%A9+de+Clermont.+Math%C3%A9matiques&rft.atitle=Th%C3%A9orie+des+%C3%A9l%C3%A9ments+saillants+d%27une+suite+d%27observations&rft.volume=Tome+8&rft.issue=2&rft.pages=7-13&rft.date=1962&rft.aulast=R%C3%A9nyi&rft.aufirst=Alfred&rft_id=http%3A%2F%2Fwww.numdam.org%2Fitem%2FASCFM_1962__8_2_7_0%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">See section 6.2 and 6.5 of <i>Concrete Mathematics</i>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="/wiki/Richard_P._Stanley" title="Richard P. Stanley">Richard P. Stanley</a>, <i>Enumerative Combinatorics, volume 1</i> (2nd ed.). Page 34 of the <a rel="nofollow" class="external text" href="http://math.mit.edu/~rstan/ec/ec1.pdf">online version</a>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdamchik1997" class="citation journal cs1">Adamchik, Victor (1997). <a rel="nofollow" class="external text" href="https://scholar.archive.org/work/7dd3uhgmr5fb5cv63sas2uqmgq">"On Stirling numbers and Euler sums"</a>. <i>Journal of Computational and Applied Mathematics</i>. <b>79</b> (1): 119–130. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0377-0427%2896%2900167-7">10.1016/S0377-0427(96)00167-7</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1437973">1437973</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+Computational+and+Applied+Mathematics&rft.atitle=On+Stirling+numbers+and+Euler+sums&rft.volume=79&rft.issue=1&rft.pages=119-130&rft.date=1997&rft_id=info%3Adoi%2F10.1016%2FS0377-0427%2896%2900167-7&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1437973%23id-name%3DMR&rft.aulast=Adamchik&rft.aufirst=Victor&rft_id=https%3A%2F%2Fscholar.archive.org%2Fwork%2F7dd3uhgmr5fb5cv63sas2uqmgq&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlajolet_and_Sedgewick1995" class="citation journal cs1">Flajolet and Sedgewick (1995). <a rel="nofollow" class="external text" href="https://hal.inria.fr/inria-00074439/file/RR-2231.pdf">"Mellin transforms and asymptotics: Finite differences and Rice's integrals"</a> <span class="cs1-format">(PDF)</span>. <i>Theoretical Computer Science</i>. <b>144</b> (1–2): 101–124. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0304-3975%2894%2900281-m">10.1016/0304-3975(94)00281-m</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theoretical+Computer+Science&rft.atitle=Mellin+transforms+and+asymptotics%3A+Finite+differences+and+Rice%27s+integrals&rft.volume=144&rft.issue=1%E2%80%932&rft.pages=101-124&rft.date=1995&rft_id=info%3Adoi%2F10.1016%2F0304-3975%2894%2900281-m&rft.au=Flajolet+and+Sedgewick&rft_id=https%3A%2F%2Fhal.inria.fr%2Finria-00074439%2Ffile%2FRR-2231.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidt2016" class="citation arxiv cs1">Schmidt, M. D. (30 October 2016). "Zeta Series Generating Function Transformations Related to Polylogarithm Functions and the <i>k</i>-Order Harmonic Numbers". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1610.09666">1610.09666</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.CO">math.CO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Zeta+Series+Generating+Function+Transformations+Related+to+Polylogarithm+Functions+and+the+k-Order+Harmonic+Numbers&rft.date=2016-10-30&rft_id=info%3Aarxiv%2F1610.09666&rft.aulast=Schmidt&rft.aufirst=M.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidt2016" class="citation arxiv cs1">Schmidt, M. D. (3 November 2016). "Zeta Series Generating Function Transformations Related to Generalized Stirling Numbers and Partial Sums of the Hurwitz Zeta Function". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1611.00957">1611.00957</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.CO">math.CO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Zeta+Series+Generating+Function+Transformations+Related+to+Generalized+Stirling+Numbers+and+Partial+Sums+of+the+Hurwitz+Zeta+Function&rft.date=2016-11-03&rft_id=info%3Aarxiv%2F1611.00957&rft.aulast=Schmidt&rft.aufirst=M.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">A table of the second-order Eulerian numbers and a synopsis of their properties is found in section 6.2 of <i>Concrete Mathematics</i>. For example, we have that <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 \sum _{k}\left\langle \!\!\left\langle {n \atop k}\right\rangle \!\!\right\rangle =(2n-1)(2n-3)\cdots 1=(2n-1)!!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow> <mo>⟨</mo> <mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k}\left\langle \!\!\left\langle {n \atop k}\right\rangle \!\!\right\rangle =(2n-1)(2n-3)\cdots 1=(2n-1)!!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29b24f23acab022515dcaa23f9ceb8811f9add55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.085ex; height:6.009ex;" alt="{\displaystyle \sum _{k}\left\langle \!\!\left\langle {n \atop k}\right\rangle \!\!\right\rangle =(2n-1)(2n-3)\cdots 1=(2n-1)!!}"></span>. These numbers also have the following combinatorial interpretation: If we form all permutations of the <a href="/wiki/Multiset" title="Multiset">multiset</a> <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 \{1,1,2,2,\ldots ,n,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,1,2,2,\ldots ,n,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32bd251e80804cbda438daefecc849991e1207c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.078ex; height:2.843ex;" alt="{\displaystyle \{1,1,2,2,\ldots ,n,n\}}"></span> with the property that all numbers between the two occurrences of <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> are greater than <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> for <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\leq k\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq k\leq n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78ec00bc2eb99b403bee93def3c12ae87f1e3c3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.965ex; height:2.343ex;" alt="{\displaystyle 1\leq k\leq n}"></span>, then <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 \left\langle \!\!\left\langle {n \atop k}\right\rangle \!\!\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle \!\!\left\langle {n \atop k}\right\rangle \!\!\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d0728889708f580ffe4b685a387737337867277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.085ex; height:4.843ex;" alt="{\displaystyle \left\langle \!\!\left\langle {n \atop k}\right\rangle \!\!\right\rangle }"></span> is the number of such permutations that have <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> ascents.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidt2016" class="citation arxiv cs1">Schmidt, M. D. (2016). "A Computer Algebra Package for Polynomial Sequence Recognition". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1609.07301">1609.07301</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.CO">math.CO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=A+Computer+Algebra+Package+for+Polynomial+Sequence+Recognition&rft.date=2016&rft_id=info%3Aarxiv%2F1609.07301&rft.aulast=Schmidt&rft.aufirst=M.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Herbert Wilf, <a rel="nofollow" class="external text" href="https://www.math.upenn.edu/~wilf/gfologyLinked2.pdf">Generatingfunctionology</a>, Section 4.6.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidt2017" class="citation journal cs1">Schmidt, M. D. (2017). <a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Schmidt/schmidt14.html">"Jacobi-Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions"</a>. <i>J. Integer Seq</i>. <b>20</b> (3). <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1610.09691">1610.09691</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Integer+Seq.&rft.atitle=Jacobi-Type+Continued+Fractions+for+the+Ordinary+Generating+Functions+of+Generalized+Factorial+Functions&rft.volume=20&rft.issue=3&rft.date=2017&rft_id=info%3Aarxiv%2F1610.09691&rft.aulast=Schmidt&rft.aufirst=M.+D.&rft_id=https%3A%2F%2Fcs.uwaterloo.ca%2Fjournals%2FJIS%2FVOL20%2FSchmidt%2Fschmidt14.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span></span> </li> <li id="cite_note-blagouch1-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-blagouch1_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-blagouch1_14-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIa._V._Blagouchine2016" class="citation journal cs1">Ia. V. Blagouchine (2016). "Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to <span class="texhtml mvar" style="font-style:italic;">π</span><sup>−1</sup>". <i>Journal of Mathematical Analysis and Applications</i>. <b>442</b> (2): 404–434. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1408.3902">1408.3902</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jmaa.2016.04.032">10.1016/j.jmaa.2016.04.032</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:119661147">119661147</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+Mathematical+Analysis+and+Applications&rft.atitle=Two+series+expansions+for+the+logarithm+of+the+gamma+function+involving+Stirling+numbers+and+containing+only+rational+coefficients+for+certain+arguments+related+to+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3E%CF%80%3C%2Fspan%3E%3Csup%3E%26minus%3B1%3C%2Fsup%3E&rft.volume=442&rft.issue=2&rft.pages=404-434&rft.date=2016&rft_id=info%3Aarxiv%2F1408.3902&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119661147%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.jmaa.2016.04.032&rft.au=Ia.+V.+Blagouchine&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1408.3902">arXiv</a></span> </li> <li id="cite_note-blag2018-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-blag2018_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlagouchine2018" class="citation journal cs1">Blagouchine, Iaroslav V. (2018). <a rel="nofollow" class="external text" href="http://math.colgate.edu/~integers/vol18a.html">"Three Notes on Ser's and Hasse's Representations for the Zeta-functions"</a>. <i>INTEGERS: The Electronic Journal of Combinatorial Number Theory</i>. <b>18A</b>: 1–45. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1606.02044">1606.02044</a></span>. <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/2016arXiv160602044B">2016arXiv160602044B</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=INTEGERS%3A+The+Electronic+Journal+of+Combinatorial+Number+Theory&rft.atitle=Three+Notes+on+Ser%27s+and+Hasse%27s+Representations+for+the+Zeta-functions&rft.volume=18A&rft.pages=1-45&rft.date=2018&rft_id=info%3Aarxiv%2F1606.02044&rft_id=info%3Abibcode%2F2016arXiv160602044B&rft.aulast=Blagouchine&rft.aufirst=Iaroslav+V.&rft_id=http%3A%2F%2Fmath.colgate.edu%2F~integers%2Fvol18a.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">See also some more interesting series representations and expansions mentioned in Connon's article: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConnon2007" class="citation arxiv cs1">Connon, D. F. (2007). "Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers (Volume I)". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0710.4022">0710.4022</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.HO">math.HO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Some+series+and+integrals+involving+the+Riemann+zeta+function%2C+binomial+coefficients+and+the+harmonic+numbers+%28Volume+I%29&rft.date=2007&rft_id=info%3Aarxiv%2F0710.4022&rft.aulast=Connon&rft.aufirst=D.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span>.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">These estimates are found in Section 26.8 of the <i>NIST Handbook of Mathematical Functions</i>.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMalenfant2011" class="citation arxiv cs1">Malenfant, Jerome (2011). "Finite, closed-form expressions for the partition function and for Euler, Bernoulli, and Stirling numbers". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1103.1585">1103.1585</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.NT">math.NT</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Finite%2C+closed-form+expressions+for+the+partition+function+and+for+Euler%2C+Bernoulli%2C+and+Stirling+numbers&rft.date=2011&rft_id=info%3Aarxiv%2F1103.1585&rft.aulast=Malenfant&rft.aufirst=Jerome&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKomatsuPita-Ruiz2018" class="citation journal cs1">Komatsu, Takao; Pita-Ruiz, Claudio (2018). <a rel="nofollow" class="external text" href="http://www.doiserbia.nb.rs/Article.aspx?ID=0354-51801811881K">"Some formulas for Bell numbers"</a>. <i>Filomat</i>. <b>32</b> (11): 3881–3889. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2298%2FFIL1811881K">10.2298/FIL1811881K</a></span>. <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/0354-5180">0354-5180</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Filomat&rft.atitle=Some+formulas+for+Bell+numbers&rft.volume=32&rft.issue=11&rft.pages=3881-3889&rft.date=2018&rft_id=info%3Adoi%2F10.2298%2FFIL1811881K&rft.issn=0354-5180&rft.aulast=Komatsu&rft.aufirst=Takao&rft.au=Pita-Ruiz%2C+Claudio&rft_id=http%3A%2F%2Fwww.doiserbia.nb.rs%2FArticle.aspx%3FID%3D0354-51801811881K&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIa._V._Blagouchine2016" class="citation journal cs1">Ia. V. Blagouchine (2016). "Expansions of generalized Euler's constants into the series of polynomials in <span class="texhtml mvar" style="font-style:italic;">π</span><sup>−2</sup> and into the formal enveloping series with rational coefficients only". <i>Journal of Number Theory</i>. <b>158</b> (2): 365–396. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jnt.2015.06.012">10.1016/j.jnt.2015.06.012</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+Number+Theory&rft.atitle=Expansions+of+generalized+Euler%27s+constants+into+the+series+of+polynomials+in+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3E%CF%80%3C%2Fspan%3E%3Csup%3E%26minus%3B2%3C%2Fsup%3E+and+into+the+formal+enveloping+series+with+rational+coefficients+only&rft.volume=158&rft.issue=2&rft.pages=365-396&rft.date=2016&rft_id=info%3Adoi%2F10.1016%2Fj.jnt.2015.06.012&rft.au=Ia.+V.+Blagouchine&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1501.00740">arXiv</a></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidt2016" class="citation arxiv cs1">Schmidt, Maxie D. (2016). "Combinatorial Identities for Generalized Stirling Numbers Expanding <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 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></span><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}"></span>-Factorial Functions and the <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 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></span><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}"></span>-Harmonic Numbers". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1611.04708">1611.04708</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.CO">math.CO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Combinatorial+Identities+for+Generalized+Stirling+Numbers+Expanding+MATH+RENDER+ERROR-Factorial+Functions+and+the+MATH+RENDER+ERROR-Harmonic+Numbers&rft.date=2016&rft_id=info%3Aarxiv%2F1611.04708&rft.aulast=Schmidt&rft.aufirst=Maxie+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidt2010" class="citation journal cs1">Schmidt, Maxie D. (2010). <a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Schmidt/multifact.html">"Generalized j-Factorial Functions, Polynomials, and Applications"</a>. <i>J. Integer Seq</i>. <b>13</b>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Integer+Seq.&rft.atitle=Generalized+j-Factorial+Functions%2C+Polynomials%2C+and+Applications&rft.volume=13&rft.date=2010&rft.aulast=Schmidt&rft.aufirst=Maxie+D.&rft_id=https%3A%2F%2Fcs.uwaterloo.ca%2Fjournals%2FJIS%2FVOL13%2FSchmidt%2Fmultifact.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><a href="/wiki/The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a></li> <li><a href="/wiki/Concrete_Mathematics" title="Concrete Mathematics">Concrete Mathematics</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFM._Abramowitz,_I._Stegun1972" class="citation book cs1">M. Abramowitz, I. Stegun, ed. (1972). "§24.1.3. Stirling Numbers of the First Kind". <i>Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</i> (9th ed.). New York: Dover. p. 824.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A724.1.3.+Stirling+Numbers+of+the+First+Kind&rft.btitle=Handbook+of+Mathematical+Functions+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&rft.place=New+York&rft.pages=824&rft.edition=9th&rft.pub=Dover&rft.date=1972&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://planetmath.org/stirlingnumbersofthefirstkind">Stirling numbers of the first kind, s(n,k)</a> at <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>..</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane_"A008275"" class="citation web cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A008275">"Sequence A008275 (Triangle read by rows of Stirling numbers of first kind)"</a>. <i>The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA008275%26%23x20%3B%28Triangle+read+by+rows+of+Stirling+numbers+of+first+kind%29&rft_id=https%3A%2F%2Foeis.org%2FA008275&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStirling+numbers+of+the+first+kind" class="Z3988"></span></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 ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol 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href="/wiki/Prime_power" title="Prime power">Prime power</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Of_the_form_a_×_2b_±_1" style="font-size:114%;margin:0 4em">Of the form <i>a</i> × 2<sup><i>b</i></sup> ± 1</div></th></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/Cullen_number" title="Cullen number">Cullen</a></li> <li><a href="/wiki/Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li> <li><a href="/wiki/Fermat_number" title="Fermat number">Fermat</a></li> <li><a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne</a></li> <li><a href="/wiki/Proth_number" class="mw-redirect" title="Proth number">Proth</a></li> <li><a href="/wiki/Thabit_number" title="Thabit number">Thabit</a></li> <li><a href="/wiki/Woodall_number" title="Woodall number">Woodall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></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/Hilbert_number" title="Hilbert number">Hilbert</a></li> <li><a href="/wiki/Idoneal_number" title="Idoneal number">Idoneal</a></li> <li><a href="/wiki/Leyland_number" title="Leyland number">Leyland</a></li> <li><a href="/wiki/Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li> <li><a href="/wiki/Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Recursion" title="Recursion">Recursively</a> defined numbers</div></th></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/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li> <li><a href="/wiki/Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li> <li><a href="/wiki/Leonardo_number" title="Leonardo number">Leonardo</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas</a></li> <li><a href="/wiki/Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li> <li><a href="/wiki/Padovan_sequence" title="Padovan sequence">Padovan</a></li> <li><a href="/wiki/Pell_number" title="Pell number">Pell</a></li> <li><a href="/wiki/Perrin_number" title="Perrin number">Perrin</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></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/Amenable_number" title="Amenable number">Amenable</a></li> <li><a href="/wiki/Congruent_number" title="Congruent number">Congruent</a></li> <li><a href="/wiki/Kn%C3%B6del_number" title="Knödel number">Knödel</a></li> <li><a href="/wiki/Riesel_number" title="Riesel number">Riesel</a></li> <li><a href="/wiki/Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></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/Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li> <li><a href="/wiki/Polite_number" title="Polite number">Polite</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li> <li><a href="/wiki/Ulam_number" title="Ulam number">Ulam</a></li> <li><a href="/wiki/Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li> <li><a href="/wiki/Centered_square_number" title="Centered square number">Centered square</a></li> <li><a href="/wiki/Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li> <li><a href="/wiki/Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal</a></li> <li><a href="/wiki/Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal</a></li> <li><a href="/wiki/Centered_octagonal_number" title="Centered octagonal number">Centered octagonal</a></li> <li><a href="/wiki/Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal</a></li> <li><a href="/wiki/Centered_decagonal_number" title="Centered decagonal number">Centered decagonal</a></li> <li><a href="/wiki/Star_number" title="Star number">Star</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polygonal_number" title="Polygonal number">non-centered</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/Triangular_number" title="Triangular number">Triangular</a></li> <li><a href="/wiki/Square_number" title="Square number">Square</a></li> <li><a href="/wiki/Square_triangular_number" title="Square triangular number">Square triangular</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal</a></li> <li><a href="/wiki/Octagonal_number" title="Octagonal number">Octagonal</a></li> <li><a href="/wiki/Nonagonal_number" title="Nonagonal number">Nonagonal</a></li> <li><a href="/wiki/Decagonal_number" title="Decagonal number">Decagonal</a></li> <li><a href="/wiki/Dodecagonal_number" title="Dodecagonal number">Dodecagonal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Centered_polyhedral_number" title="Centered polyhedral number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral</a></li> <li><a href="/wiki/Centered_cube_number" title="Centered cube number">Centered cube</a></li> <li><a href="/wiki/Centered_octahedral_number" title="Centered octahedral number">Centered octahedral</a></li> <li><a href="/wiki/Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral</a></li> <li><a href="/wiki/Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</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/Tetrahedral_number" title="Tetrahedral number">Tetrahedral</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cubic</a></li> <li><a href="/wiki/Octahedral_number" title="Octahedral number">Octahedral</a></li> <li><a href="/wiki/Dodecahedral_number" title="Dodecahedral number">Dodecahedral</a></li> <li><a href="/wiki/Icosahedral_number" title="Icosahedral number">Icosahedral</a></li> <li><a href="/wiki/Stella_octangula_number" title="Stella octangula number">Stella octangula</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Square_pyramidal_number" title="Square pyramidal number">Square pyramidal</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</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/Pentatope_number" title="Pentatope number">Pentatope</a></li> <li><a href="/wiki/Squared_triangular_number" title="Squared triangular number">Squared triangular</a></li> <li><a href="/wiki/Fourth_power" title="Fourth power">Tesseractic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_numbers" style="font-size:114%;margin:0 4em">Combinatorial numbers</div></th></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/Bell_number" title="Bell number">Bell</a></li> <li><a href="/wiki/Cake_number" title="Cake number">Cake</a></li> <li><a href="/wiki/Catalan_number" title="Catalan number">Catalan</a></li> <li><a href="/wiki/Dedekind_number" title="Dedekind number">Dedekind</a></li> <li><a href="/wiki/Delannoy_number" title="Delannoy number">Delannoy</a></li> <li><a href="/wiki/Euler_number" class="mw-redirect" title="Euler number">Euler</a></li> <li><a href="/wiki/Eulerian_number" title="Eulerian number">Eulerian</a></li> <li><a href="/wiki/Fuss%E2%80%93Catalan_number" title="Fuss–Catalan number">Fuss–Catalan</a></li> <li><a href="/wiki/Lah_number" title="Lah number">Lah</a></li> <li><a href="/wiki/Lazy_caterer%27s_sequence" title="Lazy caterer's sequence">Lazy caterer's sequence</a></li> <li><a href="/wiki/Lobb_number" title="Lobb number">Lobb</a></li> <li><a href="/wiki/Motzkin_number" title="Motzkin number">Motzkin</a></li> <li><a href="/wiki/Narayana_number" title="Narayana number">Narayana</a></li> <li><a href="/wiki/Ordered_Bell_number" title="Ordered Bell number">Ordered Bell</a></li> <li><a href="/wiki/Schr%C3%B6der_number" title="Schröder number">Schröder</a></li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Hipparchus_number" title="Schröder–Hipparchus number">Schröder–Hipparchus</a></li> <li><a class="mw-selflink selflink">Stirling first</a></li> <li><a href="/wiki/Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling second</a></li> <li><a href="/wiki/Telephone_number_(mathematics)" title="Telephone number (mathematics)">Telephone number</a></li> <li><a href="/wiki/Wedderburn%E2%80%93Etherington_number" title="Wedderburn–Etherington number">Wedderburn–Etherington</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Primes" style="font-size:114%;margin:0 4em"><a href="/wiki/Prime_number" title="Prime number">Primes</a></div></th></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/Wieferich_prime#Wieferich_numbers" title="Wieferich prime">Wieferich</a></li> <li><a href="/wiki/Wall%E2%80%93Sun%E2%80%93Sun_prime" title="Wall–Sun–Sun prime">Wall–Sun–Sun</a></li> <li><a href="/wiki/Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme prime</a></li> <li><a href="/wiki/Wilson_prime#Wilson_numbers" title="Wilson prime">Wilson</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pseudoprimes" style="font-size:114%;margin:0 4em"><a href="/wiki/Pseudoprime" title="Pseudoprime">Pseudoprimes</a></div></th></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/Carmichael_number" title="Carmichael number">Carmichael number</a></li> <li><a href="/wiki/Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a></li> <li><a href="/wiki/Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic pseudoprime</a></li> <li><a href="/wiki/Euler_pseudoprime" title="Euler pseudoprime">Euler pseudoprime</a></li> <li><a href="/wiki/Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi pseudoprime</a></li> <li><a href="/wiki/Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li> <li><a href="/wiki/Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius pseudoprime</a></li> <li><a href="/wiki/Lucas_pseudoprime" title="Lucas pseudoprime">Lucas pseudoprime</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael number</a></li> <li><a href="/wiki/Perrin_number#Perrin_primality_test" title="Perrin number">Perrin pseudoprime</a></li> <li><a href="/wiki/Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas pseudoprime</a></li> <li><a href="/wiki/Strong_pseudoprime" title="Strong pseudoprime">Strong pseudoprime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Arithmetic_functions_and_dynamics" style="font-size:114%;margin:0 4em"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Divisor_function" title="Divisor function">Divisor functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abundant_number" title="Abundant number">Abundant</a></li> <li><a href="/wiki/Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li> <li><a href="/wiki/Arithmetic_number" title="Arithmetic number">Arithmetic</a></li> <li><a href="/wiki/Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li> <li><a href="/wiki/Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li> <li><a href="/wiki/Deficient_number" title="Deficient number">Deficient</a></li> <li><a href="/wiki/Descartes_number" title="Descartes number">Descartes</a></li> <li><a href="/wiki/Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li> <li><a href="/wiki/Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li> <li><a href="/wiki/Highly_composite_number" title="Highly composite number">Highly composite</a></li> <li><a href="/wiki/Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li> <li><a href="/wiki/Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Practical_number" title="Practical number">Practical</a></li> <li><a href="/wiki/Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li> <li><a href="/wiki/Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li> <li><a href="/wiki/Refactorable_number" title="Refactorable number">Refactorable</a></li> <li><a href="/wiki/Semiperfect_number" title="Semiperfect number">Semiperfect</a></li> <li><a href="/wiki/Sublime_number" title="Sublime number">Sublime</a></li> <li><a href="/wiki/Superabundant_number" title="Superabundant number">Superabundant</a></li> <li><a href="/wiki/Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li> <li><a href="/wiki/Superperfect_number" title="Superperfect number">Superperfect</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Prime_omega_function" title="Prime omega function">Prime omega functions</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/Almost_prime" title="Almost prime">Almost prime</a></li> <li><a href="/wiki/Semiprime" title="Semiprime">Semiprime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euler%27s_totient_function" title="Euler's totient function">Euler's totient function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li> <li><a href="/wiki/Highly_totient_number" title="Highly totient number">Highly totient</a></li> <li><a href="/wiki/Noncototient" title="Noncototient">Noncototient</a></li> <li><a href="/wiki/Nontotient" title="Nontotient">Nontotient</a></li> <li><a href="/wiki/Perfect_totient_number" title="Perfect totient number">Perfect totient</a></li> <li><a href="/wiki/Sparsely_totient_number" title="Sparsely totient number">Sparsely totient</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Aliquot_sequence" title="Aliquot sequence">Aliquot sequences</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/Amicable_numbers" title="Amicable numbers">Amicable</a></li> <li><a href="/wiki/Perfect_number" title="Perfect number">Perfect</a></li> <li><a href="/wiki/Sociable_numbers" class="mw-redirect" title="Sociable numbers">Sociable</a></li> <li><a href="/wiki/Untouchable_number" title="Untouchable number">Untouchable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Primorial" title="Primorial">Primorial</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Euclid_number" title="Euclid number">Euclid</a></li> <li><a href="/wiki/Fortunate_number" title="Fortunate number">Fortunate</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_prime_factor_or_divisor_related_numbers" style="font-size:114%;margin:0 4em">Other <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> or <a href="/wiki/Divisor" title="Divisor">divisor</a> related numbers</div></th></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/Blum_integer" title="Blum integer">Blum</a></li> <li><a href="/wiki/Cyclic_number_(group_theory)" title="Cyclic number (group theory)">Cyclic</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Woods_number" title="Erdős–Woods number">Erdős–Woods</a></li> <li><a href="/wiki/Friendly_number" title="Friendly number">Friendly</a></li> <li><a href="/wiki/Giuga_number" title="Giuga number">Giuga</a></li> <li><a href="/wiki/Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li> <li><a href="/wiki/Jordan%E2%80%93P%C3%B3lya_number" title="Jordan–Pólya number">Jordan–Pólya</a></li> <li><a href="/wiki/Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael</a></li> <li><a href="/wiki/Pronic_number" title="Pronic number">Pronic</a></li> <li><a href="/wiki/Regular_number" title="Regular number">Regular</a></li> <li><a href="/wiki/Rough_number" title="Rough number">Rough</a></li> <li><a href="/wiki/Smooth_number" title="Smooth number">Smooth</a></li> <li><a href="/wiki/Sphenic_number" title="Sphenic number">Sphenic</a></li> <li><a href="/wiki/St%C3%B8rmer_number" title="Størmer number">Størmer</a></li> <li><a href="/wiki/Super-Poulet_number" title="Super-Poulet number">Super-Poulet</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Numeral_system-dependent_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Numeral_system" title="Numeral system">Numeral system</a>-dependent numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> <br />and <a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Persistence_of_a_number" title="Persistence of a number">Persistence</a> <ul><li><a href="/wiki/Additive_persistence" class="mw-redirect" title="Additive persistence">Additive</a></li> <li><a href="/wiki/Multiplicative_persistence" class="mw-redirect" title="Multiplicative persistence">Multiplicative</a></li></ul></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Digit_sum" title="Digit sum">Digit sum</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/Digit_sum" title="Digit sum">Digit sum</a></li> <li><a href="/wiki/Digital_root" title="Digital root">Digital root</a></li> <li><a href="/wiki/Self_number" title="Self number">Self</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit product</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Multiplicative_digital_root" title="Multiplicative digital root">Multiplicative digital root</a></li> <li><a href="/wiki/Sum-product_number" title="Sum-product number">Sum-product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coding-related</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/Meertens_number" title="Meertens number">Meertens</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dudeney_number" title="Dudeney number">Dudeney</a></li> <li><a href="/wiki/Factorion" title="Factorion">Factorion</a></li> <li><a href="/wiki/Kaprekar_number" title="Kaprekar number">Kaprekar</a></li> <li><a href="/wiki/Kaprekar%27s_routine" title="Kaprekar's routine">Kaprekar's constant</a></li> <li><a href="/wiki/Keith_number" title="Keith number">Keith</a></li> <li><a href="/wiki/Lychrel_number" title="Lychrel number">Lychrel</a></li> <li><a href="/wiki/Narcissistic_number" title="Narcissistic number">Narcissistic</a></li> <li><a href="/wiki/Perfect_digit-to-digit_invariant" title="Perfect digit-to-digit invariant">Perfect digit-to-digit invariant</a></li> <li><a href="/wiki/Perfect_digital_invariant" title="Perfect digital invariant">Perfect digital invariant</a> <ul><li><a href="/wiki/Happy_number" title="Happy number">Happy</a></li></ul></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">P-adic numbers</a>-related</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/Automorphic_number" title="Automorphic number">Automorphic</a> <ul><li><a href="/wiki/Trimorphic_number" class="mw-redirect" title="Trimorphic number">Trimorphic</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_digit" title="Numerical digit">Digit</a>-composition related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Palindromic_number" title="Palindromic number">Palindromic</a></li> <li><a href="/wiki/Pandigital_number" title="Pandigital number">Pandigital</a></li> <li><a href="/wiki/Repdigit" title="Repdigit">Repdigit</a></li> <li><a href="/wiki/Repunit" title="Repunit">Repunit</a></li> <li><a href="/wiki/Self-descriptive_number" title="Self-descriptive number">Self-descriptive</a></li> <li><a href="/wiki/Smarandache%E2%80%93Wellin_number" title="Smarandache–Wellin number">Smarandache–Wellin</a></li> <li><a href="/wiki/Undulating_number" title="Undulating number">Undulating</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit-<a href="/wiki/Permutation" title="Permutation">permutation</a> related</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/Cyclic_number" title="Cyclic number">Cyclic</a></li> <li><a href="/wiki/Digit-reassembly_number" title="Digit-reassembly number">Digit-reassembly</a></li> <li><a href="/wiki/Parasitic_number" title="Parasitic number">Parasitic</a></li> <li><a href="/wiki/Primeval_number" title="Primeval number">Primeval</a></li> <li><a href="/wiki/Transposable_integer" title="Transposable integer">Transposable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisor-related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Equidigital_number" title="Equidigital number">Equidigital</a></li> <li><a href="/wiki/Extravagant_number" title="Extravagant number">Extravagant</a></li> <li><a href="/wiki/Frugal_number" title="Frugal number">Frugal</a></li> <li><a href="/wiki/Harshad_number" title="Harshad number">Harshad</a></li> <li><a href="/wiki/Polydivisible_number" title="Polydivisible number">Polydivisible</a></li> <li><a href="/wiki/Smith_number" title="Smith number">Smith</a></li> <li><a href="/wiki/Vampire_number" title="Vampire number">Vampire</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</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/Friedman_number" title="Friedman number">Friedman</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Binary_numbers" style="font-size:114%;margin:0 4em"><a href="/wiki/Binary_number" title="Binary number">Binary numbers</a></div></th></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/Evil_number" title="Evil number">Evil</a></li> <li><a href="/wiki/Odious_number" title="Odious number">Odious</a></li> <li><a href="/wiki/Pernicious_number" title="Pernicious number">Pernicious</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Generated_via_a_sieve" style="font-size:114%;margin:0 4em">Generated via a <a href="/wiki/Sieve_theory" title="Sieve theory">sieve</a></div></th></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/Lucky_number" title="Lucky number">Lucky</a></li> <li><a href="/wiki/Generation_of_primes" title="Generation of primes">Prime</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Sorting_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Sorting_algorithm" title="Sorting algorithm">Sorting</a> related</div></th></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/Pancake_sorting" title="Pancake sorting">Pancake number</a></li> <li><a href="/wiki/Sorting_number" title="Sorting number">Sorting number</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Natural_language_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Natural_language" title="Natural language">Natural language</a> related</div></th></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/Aronson%27s_sequence" title="Aronson's sequence">Aronson's sequence</a></li> <li><a href="/wiki/Ban_number" title="Ban number">Ban</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graphemics_related" style="font-size:114%;margin:0 4em"><a href="/wiki/Graphemics" title="Graphemics">Graphemics</a> related</div></th></tr><tr><td colspan="2" 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Stirling numbers of the first kind - Wikipedia