Double factorial - Wikipedia

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div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style> <div role="note" class="hatnote navigation-not-searchable"> The double factorial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!!} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511717d541dba5357928e8d8631f1b4d4f8d5b31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.688ex; height:2.176ex;" alt="{\displaystyle n!!}"> is not the same as applying the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Factorial">factorial</a> function twice <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n!)!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> ! </mo> <mo stretchy="false"> ) </mo> <mo> ! </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (n!)!} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beef2ffd8d0ceb0f080206f0da63d9c6b9d6070b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.498ex; height:2.843ex;" alt="{\displaystyle (n!)!}">. </div> In <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematics">mathematics</a>, the double factorial of a number n, denoted by n‼, is the product of all the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Positive_integer?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Positive integer">positive integers</a> up to n that have the same <a href="https://en-m-wikipedia-org.translate.goog/wiki/Parity_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Parity (mathematics)">parity</a> (odd or even) as n.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-callan-1">[1]</a> That is, <figure typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Chord_diagrams_K6_matchings.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Chord_diagrams_K6_matchings.svg/360px-Chord_diagrams_K6_matchings.svg.png" decoding="async" width="360" height="213" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/0/06/Chord_diagrams_K6_matchings.svg/540px-Chord_diagrams_K6_matchings.svg.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/0/06/Chord_diagrams_K6_matchings.svg/720px-Chord_diagrams_K6_matchings.svg.png 2x" data-file-width="972" data-file-height="576"></a> <figcaption> The fifteen different <a href="https://en-m-wikipedia-org.translate.goog/wiki/Chord_diagram_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Chord diagram (mathematics)">chord diagrams</a> on six points, or equivalently the fifteen different <a href="https://en-m-wikipedia-org.translate.goog/wiki/Perfect_matching?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Perfect matching">perfect matchings</a> on a six-vertex <a href="https://en-m-wikipedia-org.translate.goog/wiki/Complete_graph?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Complete graph">complete graph</a>. These are counted by the double factorial 15 = (6 − 1)‼. </figcaption> </figure> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!!=\prod _{k=0}^{\left\lceil {\frac {n}{2}}\right\rceil -1}(n-2k)=n(n-2)(n-4)\cdots .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> ! </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"> <mrow> <mo> ⌈ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ⌉ </mo> </mrow> <mo> − </mo> <mn> 1 </mn> </mrow> </munderover> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 2 </mn> <mi> k </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> n </mi> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 4 </mn> <mo stretchy="false"> ) </mo> <mo> ⋯ </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!!=\prod _{k=0}^{\left\lceil {\frac {n}{2}}\right\rceil -1}(n-2k)=n(n-2)(n-4)\cdots .} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd4bb531283bcca76191054e0a1dfc761bc9265e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.729ex; height:9.009ex;" alt="{\displaystyle n!!=\prod _{k=0}^{\left\lceil {\frac {n}{2}}\right\rceil -1}(n-2k)=n(n-2)(n-4)\cdots .}"> Restated, this says that for even n, the double factorial<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-2">[2]</a> is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!!=\prod _{k=1}^{\frac {n}{2}}(2k)=n(n-2)(n-4)\cdots 4\cdot 2\,,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <munderover> <mo> ∏ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> </munderover> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> n </mi> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 4 </mn> <mo stretchy="false"> ) </mo> <mo> ⋯ </mo> <mn> 4 </mn> <mo> ⋅ </mo> <mn> 2 </mn> <mspace width="thinmathspace"></mspace> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!!=\prod _{k=1}^{\frac {n}{2}}(2k)=n(n-2)(n-4)\cdots 4\cdot 2\,,} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2c6b3ce11a2addbe78d4bdf74e3f030df3296d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.382ex; height:8.343ex;" alt="{\displaystyle n!!=\prod _{k=1}^{\frac {n}{2}}(2k)=n(n-2)(n-4)\cdots 4\cdot 2\,,}"> while for odd n it is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!!=\prod _{k=1}^{\frac {n+1}{2}}(2k-1)=n(n-2)(n-4)\cdots 3\cdot 1\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <munderover> <mo> ∏ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> </munderover> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> n </mi> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 4 </mn> <mo stretchy="false"> ) </mo> <mo> ⋯ </mo> <mn> 3 </mn> <mo> ⋅ </mo> <mn> 1 </mn> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!!=\prod _{k=1}^{\frac {n+1}{2}}(2k-1)=n(n-2)(n-4)\cdots 3\cdot 1\,.} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc7e4a63cca4a1cc136e5b72bd2de4db575d8c83" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.757ex; height:8.676ex;" alt="{\displaystyle n!!=\prod _{k=1}^{\frac {n+1}{2}}(2k-1)=n(n-2)(n-4)\cdots 3\cdot 1\,.}"> For example, 9‼ = 9 × 7 × 5 × 3 × 1 = 945. The zero double factorial 0‼ = 1 as an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Empty_product?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Empty product">empty product</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:0-3">[3]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-4">[4]</a> The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sequence?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sequence">sequence</a> of double factorials for even n = 0, 2, 4, 6, 8,... starts as <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style> <div class="block-indent"> 1, 2, 8, 48, 384, 3840, 46080, 645120, ... (sequence <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://oeis.org/A000165" class="extiw" title="oeis:A000165">A000165</a> in the <a href="https://en-m-wikipedia-org.translate.goog/wiki/On-Line_Encyclopedia_of_Integer_Sequences?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </div> The sequence of double factorials for odd n = 1, 3, 5, 7, 9,... starts as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573"> <div class="block-indent"> 1, 3, 15, 105, 945, 10395, 135135, ... (sequence <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://oeis.org/A001147" class="extiw" title="oeis:A001147">A001147</a> in the <a href="https://en-m-wikipedia-org.translate.goog/wiki/On-Line_Encyclopedia_of_Integer_Sequences?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>) </div> The term odd factorial is sometimes used for the double factorial of an odd number.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-5">[5]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-6">[6]</a> The term semifactorial is also used by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Donald_Knuth?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Donald Knuth">Knuth</a> as a synonym of double factorial.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-7">[7]</a> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="en" dir="ltr"> <h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#History_and_usage">1 History and usage</a></li> <li class="toclevel-1 tocsection-2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Relation_to_the_factorial">2 Relation to the factorial</a></li> <li class="toclevel-1 tocsection-3"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Applications_in_enumerative_combinatorics">3 Applications in enumerative combinatorics</a></li> <li class="toclevel-1 tocsection-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Asymptotics">4 Asymptotics</a></li> <li class="toclevel-1 tocsection-5"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Extensions">5 Extensions</a> <ul> <li class="toclevel-2 tocsection-6"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Negative_arguments">5.1 Negative arguments</a></li> <li class="toclevel-2 tocsection-7"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Complex_arguments">5.2 Complex arguments</a></li> </ul></li> <li class="toclevel-1 tocsection-8"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Additional_identities">6 Additional identities</a></li> <li class="toclevel-1 tocsection-9"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Generalizations">7 Generalizations</a> <ul> <li class="toclevel-2 tocsection-10"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Definitions">7.1 Definitions</a></li> <li class="toclevel-2 tocsection-11"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Alternative_extension_of_the_multifactorial">7.2 Alternative extension of the multifactorial</a></li> <li class="toclevel-2 tocsection-12"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Generalized_Stirling_numbers_expanding_the_multifactorial_functions">7.3 Generalized Stirling numbers expanding the multifactorial functions</a></li> <li class="toclevel-2 tocsection-13"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Exact_finite_sums_involving_multiple_factorial_functions">7.4 Exact finite sums involving multiple factorial functions</a></li> </ul></li> <li class="toclevel-1 tocsection-14"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#References">8 References</a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <h2 id="History_and_usage">History and usage</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Double_factorial&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: History and usage" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> In a 1902 paper, the physicist <a href="https://en-m-wikipedia-org.translate.goog/wiki/Arthur_Schuster?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Arthur Schuster">Arthur Schuster</a> wrote:<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-8">[8]</a> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style> <blockquote class="templatequote"> The symbolical representation of the results of this paper is much facilitated by the introduction of a separate symbol for the product of alternate factors, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\cdot n-2\cdot n-4\cdots 1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ⋅ </mo> <mi> n </mi> <mo> − </mo> <mn> 2 </mn> <mo> ⋅ </mo> <mi> n </mi> <mo> − </mo> <mn> 4 </mn> <mo> ⋯ </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n\cdot n-2\cdot n-4\cdots 1} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83a1ebea406396177824208a39efb2087d3ecedb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.208ex; height:2.343ex;" alt="{\displaystyle n\cdot n-2\cdot n-4\cdots 1}"> </noscript><span class="lazy-image-placeholder" style="width: 20.208ex;height: 2.343ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83a1ebea406396177824208a39efb2087d3ecedb" data-alt="{\displaystyle n\cdot n-2\cdot n-4\cdots 1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , if <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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  be odd, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\cdot n-2\cdots 2}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ⋅ </mo> <mi> n </mi> <mo> − </mo> <mn> 2 </mn> <mo> ⋯ </mo> <mn> 2 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n\cdot n-2\cdots 2} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17585cc0b0ef9c4dae965be4370c5c27a719dafd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.131ex; height:2.343ex;" alt="{\displaystyle n\cdot n-2\cdots 2}"> </noscript><span class="lazy-image-placeholder" style="width: 13.131ex;height: 2.343ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17585cc0b0ef9c4dae965be4370c5c27a719dafd" data-alt="{\displaystyle n\cdot n-2\cdots 2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  if <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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  be odd [sic]. I propose to write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!!} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511717d541dba5357928e8d8631f1b4d4f8d5b31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.688ex; height:2.176ex;" alt="{\displaystyle n!!}"> </noscript><span class="lazy-image-placeholder" style="width: 2.688ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511717d541dba5357928e8d8631f1b4d4f8d5b31" data-alt="{\displaystyle n!!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  for such products, and if a name be required for the product to call it the "alternate factorial" or the "double factorial". </blockquote> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFMeserve1948">Meserve (1948)</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-meserve-9">[9]</a> states that the double factorial was originally introduced in order to simplify the expression of certain <a href="https://en-m-wikipedia-org.translate.goog/wiki/List_of_integrals_of_trigonometric_functions?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="List of integrals of trigonometric functions">trigonometric integrals</a> that arise in the derivation of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wallis_product?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wallis product">Wallis product</a>. Double factorials also arise in expressing the volume of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ball_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ball (mathematics)">hyperball</a> and surface area of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/N-sphere?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="N-sphere">hypersphere</a>, and they have many applications in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Enumerative_combinatorics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Enumerative combinatorics">enumerative combinatorics</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-callan-1">[1]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-dm93-10">[10]</a> They occur in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Student%27s_t-distribution?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Student's t-distribution">Student's t-distribution</a> (1908), though <a href="https://en-m-wikipedia-org.translate.goog/wiki/William_Sealy_Gosset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="William Sealy Gosset">Gosset</a> did not use the double exclamation point notation. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <h2 id="Relation_to_the_factorial">Relation to the factorial</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Double_factorial&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Relation to the factorial" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> Because the double factorial only involves about half the factors of the ordinary <a href="https://en-m-wikipedia-org.translate.goog/wiki/Factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Factorial">factorial</a>, its value is not substantially larger than the square root of the factorial n!, and it is much smaller than the iterated factorial (n!)!. The factorial of a positive n may be written as the product of two double factorials:<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:0-3">[3]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!=n!!\cdot (n-1)!!\,,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> = </mo> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mspace width="thinmathspace"></mspace> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!=n!!\cdot (n-1)!!\,,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ced9d3582cacf1e332848bb63f115cfc3f2ccf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.042ex; height:2.843ex;" alt="{\displaystyle n!=n!!\cdot (n-1)!!\,,}"> </noscript><span class="lazy-image-placeholder" style="width: 19.042ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ced9d3582cacf1e332848bb63f115cfc3f2ccf" data-alt="{\displaystyle n!=n!!\cdot (n-1)!!\,,}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  and therefore <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!!={\frac {n!}{(n-1)!!}}={\frac {(n+1)!}{(n+1)!!}}\,,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mrow> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> </mrow> <mrow> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!!={\frac {n!}{(n-1)!!}}={\frac {(n+1)!}{(n+1)!!}}\,,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8f9bc544e3e74be89c6b0f0ee53fb0fea80c04" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.593ex; height:6.509ex;" alt="{\displaystyle n!!={\frac {n!}{(n-1)!!}}={\frac {(n+1)!}{(n+1)!!}}\,,}"> </noscript><span class="lazy-image-placeholder" style="width: 28.593ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8f9bc544e3e74be89c6b0f0ee53fb0fea80c04" data-alt="{\displaystyle n!!={\frac {n!}{(n-1)!!}}={\frac {(n+1)!}{(n+1)!!}}\,,}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where the denominator cancels the unwanted factors in the numerator. (The last form also applies when n = 0.) For an even non-negative integer n = 2k with k ≥ 0, the double factorial may be expressed as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2k)!!=2^{k}k!\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mi> k </mi> <mo> ! </mo> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (2k)!!=2^{k}k!\,.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/581c58950469fd9cb703197776c625debcae5067" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.718ex; height:3.176ex;" alt="{\displaystyle (2k)!!=2^{k}k!\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 13.718ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/581c58950469fd9cb703197776c625debcae5067" data-alt="{\displaystyle (2k)!!=2^{k}k!\,.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  For odd n = 2k − 1 with k ≥ 1, combining the two previous formulas yields <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2k-1)!!={\frac {(2k)!}{2^{k}k!}}={\frac {(2k-1)!}{2^{k-1}(k-1)!}}\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo stretchy="false"> ) </mo> <mo> ! </mo> </mrow> <mrow> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> </mrow> <mrow> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (2k-1)!!={\frac {(2k)!}{2^{k}k!}}={\frac {(2k-1)!}{2^{k-1}(k-1)!}}\,.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae8722d4f42da84334ed4e3698318f1f65dbffb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.234ex; height:6.843ex;" alt="{\displaystyle (2k-1)!!={\frac {(2k)!}{2^{k}k!}}={\frac {(2k-1)!}{2^{k-1}(k-1)!}}\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 35.234ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae8722d4f42da84334ed4e3698318f1f65dbffb" data-alt="{\displaystyle (2k-1)!!={\frac {(2k)!}{2^{k}k!}}={\frac {(2k-1)!}{2^{k-1}(k-1)!}}\,.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  For an odd positive integer n = 2k − 1 with k ≥ 1, the double factorial may be expressed in terms of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Permutation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Permutations_without_repetitions" title="Permutation">k-permutations of 2k</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-callan-1">[1]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-gq12-11">[11]</a> or a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Falling_and_rising_factorials?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Falling and rising factorials">falling factorial</a> as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2k-1)!!={\frac {_{2k}P_{k}}{2^{k}}}={\frac {(2k)^{\underline {k}}}{2^{k}}}\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> k </mi> </mrow> </msub> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi> k </mi> <mo> _ </mo> </munder> </mrow> </msup> </mrow> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (2k-1)!!={\frac {_{2k}P_{k}}{2^{k}}}={\frac {(2k)^{\underline {k}}}{2^{k}}}\,.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f2528ee6c64244989ae128c502ea12513fa921f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.148ex; height:6.343ex;" alt="{\displaystyle (2k-1)!!={\frac {_{2k}P_{k}}{2^{k}}}={\frac {(2k)^{\underline {k}}}{2^{k}}}\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 28.148ex;height: 6.343ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f2528ee6c64244989ae128c502ea12513fa921f" data-alt="{\displaystyle (2k-1)!!={\frac {_{2k}P_{k}}{2^{k}}}={\frac {(2k)^{\underline {k}}}{2^{k}}}\,.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <h2 id="Applications_in_enumerative_combinatorics">Applications in enumerative combinatorics</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Double_factorial&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Applications in enumerative combinatorics" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <figure typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Unordered_binary_trees_with_4_leaves.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Unordered_binary_trees_with_4_leaves.svg/300px-Unordered_binary_trees_with_4_leaves.svg.png" decoding="async" width="300" height="210" class="mw-file-element" data-file-width="630" data-file-height="441"> </noscript><span class="lazy-image-placeholder" style="width: 300px;height: 210px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Unordered_binary_trees_with_4_leaves.svg/300px-Unordered_binary_trees_with_4_leaves.svg.png" data-width="300" data-height="210" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Unordered_binary_trees_with_4_leaves.svg/450px-Unordered_binary_trees_with_4_leaves.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Unordered_binary_trees_with_4_leaves.svg/600px-Unordered_binary_trees_with_4_leaves.svg.png 2x" data-class="mw-file-element"> </a> <figcaption> The fifteen different <a href="https://en-m-wikipedia-org.translate.goog/wiki/Rooted_binary_tree?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Rooted binary tree">rooted binary trees</a> (with unordered children) on a set of four labeled leaves, illustrating 15 = (2 × 4 − 3)‼ (see article text). </figcaption> </figure> Double factorials are motivated by the fact that they occur frequently in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Enumerative_combinatorics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Enumerative combinatorics">enumerative combinatorics</a> and other settings. For instance, n‼ for odd values of n counts <ul> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Perfect_matching?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Perfect matching">Perfect matchings</a> of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Complete_graph?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Complete graph">complete graph</a> Kn + 1 for odd n. In such a graph, any single vertex v has n possible choices of vertex that it can be matched to, and once this choice is made the remaining problem is one of selecting a perfect matching in a complete graph with two fewer vertices. For instance, a complete graph with four vertices a, b, c, and d has three perfect matchings: ab and cd, ac and bd, and ad and bc.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-callan-1">[1]</a> Perfect matchings may be described in several other equivalent ways, including <a href="https://en-m-wikipedia-org.translate.goog/wiki/Involution_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Involution (mathematics)">involutions</a> without fixed points on a set of n + 1 items (<a href="https://en-m-wikipedia-org.translate.goog/wiki/Permutations?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Permutations">permutations</a> in which each cycle is a pair)<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-callan-1">[1]</a> or <a href="https://en-m-wikipedia-org.translate.goog/wiki/Chord_diagram_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Chord diagram (mathematics)">chord diagrams</a> (sets of chords of a set of n + 1 points evenly spaced on a circle such that each point is the endpoint of exactly one chord, also called <a href="https://en-m-wikipedia-org.translate.goog/wiki/Richard_Brauer?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Richard Brauer">Brauer</a> diagrams).<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-dm93-10">[10]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-12">[12]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-13">[13]</a> The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are instead given by the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Telephone_number_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Telephone number (mathematics)">telephone numbers</a>, which may be expressed as a summation involving double factorials.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-14">[14]</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling_permutation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Stirling permutation">Stirling permutations</a>, permutations of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Multiset">multiset</a> of numbers 1, 1, 2, 2, ..., k, k in which each pair of equal numbers is separated only by larger numbers, where k = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠n + 1/2⁠. The two copies of k must be adjacent; removing them from the permutation leaves a permutation in which the maximum element is k − 1, with n positions into which the adjacent pair of k values may be placed. From this recursive construction, a proof that the Stirling permutations are counted by the double permutations follows by induction.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-callan-1">[1]</a> Alternatively, instead of the restriction that values between a pair may be larger than it, one may also consider the permutations of this multiset in which the first copies of each pair appear in sorted order; such a permutation defines a matching on the 2k positions of the permutation, so again the number of permutations may be counted by the double permutations.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-dm93-10">[10]</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Heap_(data_structure)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Heap (data structure)">Heap-ordered trees</a>, trees with k + 1 nodes labeled 0, 1, 2, ... k, such that the root of the tree has label 0, each other node has a larger label than its parent, and such that the children of each node have a fixed ordering. An <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euler_tour_technique?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euler tour technique">Euler tour</a> of the tree (with doubled edges) gives a Stirling permutation, and every Stirling permutation represents a tree in this way.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-callan-1">[1]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-15">[15]</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Unrooted_binary_tree?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Unrooted binary tree">Unrooted binary trees</a> with <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠n + 5/2⁠ labeled leaves. Each such tree may be formed from a tree with one fewer leaf, by subdividing one of the n tree edges and making the new vertex be the parent of a new leaf.</li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Rooted_binary_tree?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Rooted binary tree">Rooted binary trees</a> with <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠n + 3/2⁠ labeled leaves. This case is similar to the unrooted case, but the number of edges that can be subdivided is even, and in addition to subdividing an edge it is possible to add a node to a tree with one fewer leaf by adding a new root whose two children are the smaller tree and the new leaf.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-callan-1">[1]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-dm93-10">[10]</a></li> </ul> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFCallan2009">Callan (2009)</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFDaleMoon1993">Dale & Moon (1993)</a> list several additional objects with the same <a href="https://en-m-wikipedia-org.translate.goog/wiki/Combinatorial_class?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Combinatorial class">counting sequence</a>, including "trapezoidal words" (<a href="https://en-m-wikipedia-org.translate.goog/wiki/Numeral_system?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Numeral system">numerals</a> in a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mixed_radix?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mixed radix">mixed radix</a> system with increasing odd radixes), height-labeled <a href="https://en-m-wikipedia-org.translate.goog/wiki/Dyck_path?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Dyck path">Dyck paths</a>, height-labeled ordered trees, "overhang paths", and certain vectors describing the lowest-numbered leaf descendant of each node in a rooted binary tree. For <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bijective_proof?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bijective proof">bijective proofs</a> that some of these objects are equinumerous, see <a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFRubey2008">Rubey (2008)</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFMarshMartin2011">Marsh & Martin (2011)</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-16">[16]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-17">[17]</a> The even double factorials give the numbers of elements of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hyperoctahedral_group?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hyperoctahedral group">hyperoctahedral groups</a> (signed permutations or symmetries of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hypercube?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hypercube">hypercube</a>) </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <h2 id="Asymptotics">Asymptotics</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Double_factorial&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Asymptotics" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling%27s_approximation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Stirling's approximation">Stirling's approximation</a> for the factorial can be used to derive an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Asymptotic_analysis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Asymptotic analysis">asymptotic equivalent</a> for the double factorial. In particular, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> ∼ </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> <mi> π </mi> <mi> n </mi> </msqrt> </mrow> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> n </mi> <mi> e </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe20ccef4b13b2fc2b79b752fb595da6d855de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.837ex; height:4.843ex;" alt="{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n},}"> </noscript><span class="lazy-image-placeholder" style="width: 17.837ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe20ccef4b13b2fc2b79b752fb595da6d855de2" data-alt="{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  one has as <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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  tends to infinity that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!!\sim {\begin{cases}\displaystyle {\sqrt {\pi n}}\left({\frac {n}{e}}\right)^{n/2}&{\text{if }}n{\text{ is even}},\\[5pt]\displaystyle {\sqrt {2n}}\left({\frac {n}{e}}\right)^{n/2}&{\text{if }}n{\text{ is odd}}.\end{cases}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> <mo> ∼ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> { </mo> <mtable columnalign="left left" rowspacing="0.7em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi> π </mi> <mi> n </mi> </msqrt> </mrow> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> n </mi> <mi> e </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> </mstyle> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> if  </mtext> </mrow> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>  is even </mtext> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> <mi> n </mi> </msqrt> </mrow> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> n </mi> <mi> e </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> </mstyle> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> if  </mtext> </mrow> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>  is odd </mtext> </mrow> <mo> . </mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!!\sim {\begin{cases}\displaystyle {\sqrt {\pi n}}\left({\frac {n}{e}}\right)^{n/2}&{\text{if }}n{\text{ is even}},\\[5pt]\displaystyle {\sqrt {2n}}\left({\frac {n}{e}}\right)^{n/2}&{\text{if }}n{\text{ is odd}}.\end{cases}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1665e6a09f9b9a28da13ed08f17b66361aec1364" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:34.748ex; height:11.843ex;" alt="{\displaystyle n!!\sim {\begin{cases}\displaystyle {\sqrt {\pi n}}\left({\frac {n}{e}}\right)^{n/2}&{\text{if }}n{\text{ is even}},\\[5pt]\displaystyle {\sqrt {2n}}\left({\frac {n}{e}}\right)^{n/2}&{\text{if }}n{\text{ is odd}}.\end{cases}}}"> </noscript><span class="lazy-image-placeholder" style="width: 34.748ex;height: 11.843ex;vertical-align: -5.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1665e6a09f9b9a28da13ed08f17b66361aec1364" data-alt="{\displaystyle n!!\sim {\begin{cases}\displaystyle {\sqrt {\pi n}}\left({\frac {n}{e}}\right)^{n/2}&{\text{if }}n{\text{ is even}},\\[5pt]\displaystyle {\sqrt {2n}}\left({\frac {n}{e}}\right)^{n/2}&{\text{if }}n{\text{ is odd}}.\end{cases}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <h2 id="Extensions">Extensions</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Double_factorial&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Extensions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <div class="mw-heading mw-heading3"> <h3 id="Negative_arguments">Negative arguments</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Double_factorial&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Negative arguments" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The ordinary factorial, when extended to the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gamma_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gamma function">gamma function</a>, has a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Pole_(complex_analysis)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Pole (complex analysis)">pole</a> at each negative integer, preventing the factorial from being defined at these numbers. However, the double factorial of odd numbers may be extended to any negative odd integer argument by inverting its <a href="https://en-m-wikipedia-org.translate.goog/wiki/Recurrence_relation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Recurrence relation">recurrence relation</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!!=n\times (n-2)!!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <mi> n </mi> <mo> × </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!!=n\times (n-2)!!} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5981330b4c21bfe4834d3afff0b140e22b2a017" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.523ex; height:2.843ex;" alt="{\displaystyle n!!=n\times (n-2)!!}"> </noscript><span class="lazy-image-placeholder" style="width: 18.523ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5981330b4c21bfe4834d3afff0b140e22b2a017" data-alt="{\displaystyle n!!=n\times (n-2)!!}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  to give <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!!={\frac {(n+2)!!}{n+2}}\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!!={\frac {(n+2)!!}{n+2}}\,.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/662214e367d284c0405ef36d9493530dd7116006" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.157ex; height:5.843ex;" alt="{\displaystyle n!!={\frac {(n+2)!!}{n+2}}\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 16.157ex;height: 5.843ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/662214e367d284c0405ef36d9493530dd7116006" data-alt="{\displaystyle n!!={\frac {(n+2)!!}{n+2}}\,.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Using this inverted recurrence, (−1)!! = 1, (−3)!! = −1, and (−5)!! =  <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/3⁠; negative odd numbers with greater magnitude have fractional double factorials.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-callan-1">[1]</a> In particular, when n is an odd number, this gives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-n)!!\times n!!=(-1)^{\frac {n-1}{2}}\times n\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mo> − </mo> <mi> n </mi> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mo> × </mo> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 1 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> × </mo> <mi> n </mi> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (-n)!!\times n!!=(-1)^{\frac {n-1}{2}}\times n\,.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3910ca161950ab98637d93ea3308bc037d8a763d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.556ex; height:4.176ex;" alt="{\displaystyle (-n)!!\times n!!=(-1)^{\frac {n-1}{2}}\times n\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 28.556ex;height: 4.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3910ca161950ab98637d93ea3308bc037d8a763d" data-alt="{\displaystyle (-n)!!\times n!!=(-1)^{\frac {n-1}{2}}\times n\,.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  <div class="mw-heading mw-heading3"> <h3 id="Complex_arguments">Complex arguments</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Double_factorial&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Complex arguments" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Disregarding the above definition of n!! for even values of n, the double factorial for odd integers can be extended to most real and complex numbers z by noting that when z is a positive odd integer then<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-18">[18]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-19">[19]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}z!!&=z(z-2)\cdots 5\cdot 3\\[3mu]&=2^{\frac {z-1}{2}}\left({\frac {z}{2}}\right)\left({\frac {z-2}{2}}\right)\cdots \left({\frac {5}{2}}\right)\left({\frac {3}{2}}\right)\\[5mu]&=2^{\frac {z-1}{2}}{\frac {\Gamma \left({\tfrac {z}{2}}+1\right)}{\Gamma \left({\tfrac {1}{2}}+1\right)}}\\[5mu]&={\sqrt {\frac {2}{\pi }}}2^{\frac {z}{2}}\Gamma \left({\tfrac {z}{2}}+1\right)\,,\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.467em 0.578em 0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi> z </mi> <mo> ! </mo> <mo> ! </mo> </mtd> <mtd> <mi></mi> <mo> = </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo> − </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> <mo> ⋯ </mo> <mn> 5 </mn> <mo> ⋅ </mo> <mn> 3 </mn> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> z </mi> <mo> − </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> </msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> z </mi> <mo> − </mo> <mn> 2 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⋯ </mo> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 5 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> z </mi> <mo> − </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> Γ </mi> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi mathvariant="normal"> Γ </mi> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> </msqrt> </mrow> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mi mathvariant="normal"> Γ </mi> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mspace width="thinmathspace"></mspace> <mo> , </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}z!!&=z(z-2)\cdots 5\cdot 3\\[3mu]&=2^{\frac {z-1}{2}}\left({\frac {z}{2}}\right)\left({\frac {z-2}{2}}\right)\cdots \left({\frac {5}{2}}\right)\left({\frac {3}{2}}\right)\\[5mu]&=2^{\frac {z-1}{2}}{\frac {\Gamma \left({\tfrac {z}{2}}+1\right)}{\Gamma \left({\tfrac {1}{2}}+1\right)}}\\[5mu]&={\sqrt {\frac {2}{\pi }}}2^{\frac {z}{2}}\Gamma \left({\tfrac {z}{2}}+1\right)\,,\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1a14e8300bf258355a029306178b16076cce5bb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.005ex; width:40.413ex; height:25.176ex;" alt="{\displaystyle {\begin{aligned}z!!&=z(z-2)\cdots 5\cdot 3\\[3mu]&=2^{\frac {z-1}{2}}\left({\frac {z}{2}}\right)\left({\frac {z-2}{2}}\right)\cdots \left({\frac {5}{2}}\right)\left({\frac {3}{2}}\right)\\[5mu]&=2^{\frac {z-1}{2}}{\frac {\Gamma \left({\tfrac {z}{2}}+1\right)}{\Gamma \left({\tfrac {1}{2}}+1\right)}}\\[5mu]&={\sqrt {\frac {2}{\pi }}}2^{\frac {z}{2}}\Gamma \left({\tfrac {z}{2}}+1\right)\,,\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 40.413ex;height: 25.176ex;vertical-align: -12.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1a14e8300bf258355a029306178b16076cce5bb" data-alt="{\displaystyle {\begin{aligned}z!!&=z(z-2)\cdots 5\cdot 3\\[3mu]&=2^{\frac {z-1}{2}}\left({\frac {z}{2}}\right)\left({\frac {z-2}{2}}\right)\cdots \left({\frac {5}{2}}\right)\left({\frac {3}{2}}\right)\\[5mu]&=2^{\frac {z-1}{2}}{\frac {\Gamma \left({\tfrac {z}{2}}+1\right)}{\Gamma \left({\tfrac {1}{2}}+1\right)}}\\[5mu]&={\sqrt {\frac {2}{\pi }}}2^{\frac {z}{2}}\Gamma \left({\tfrac {z}{2}}+1\right)\,,\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <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> <noscript> <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)}"> </noscript><span class="lazy-image-placeholder" style="width: 4.35ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11ca17f880240539116aac7e6326909299e2a080" data-alt="{\displaystyle \Gamma (z)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gamma_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gamma function">gamma function</a>. The final expression is defined for all complex numbers except the negative even integers and satisfies (z + 2)!! = (z + 2) · z!! everywhere it is defined. As with the gamma function that extends the ordinary factorial function, this double factorial function is <a href="https://en-m-wikipedia-org.translate.goog/wiki/Logarithmically_convex?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Logarithmically convex">logarithmically convex</a> in the sense of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bohr%E2%80%93Mollerup_theorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bohr–Mollerup theorem">Bohr–Mollerup theorem</a>. Asymptotically, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n!!\sim {\sqrt {2n^{n+1}e^{-n}}}\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> <mo> ∼ </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> n </mi> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\textstyle n!!\sim {\sqrt {2n^{n+1}e^{-n}}}\,.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acd30e60ef8585bfc10f9e569fbae5420fc3ec38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.213ex; height:3.009ex;" alt="{\textstyle n!!\sim {\sqrt {2n^{n+1}e^{-n}}}\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 18.213ex;height: 3.009ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acd30e60ef8585bfc10f9e569fbae5420fc3ec38" data-alt="{\textstyle n!!\sim {\sqrt {2n^{n+1}e^{-n}}}\,.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  The generalized formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\frac {2}{\pi }}}2^{\frac {z}{2}}\Gamma \left({\tfrac {z}{2}}+1\right)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> </msqrt> </mrow> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mi mathvariant="normal"> Γ </mi> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\sqrt {\frac {2}{\pi }}}2^{\frac {z}{2}}\Gamma \left({\tfrac {z}{2}}+1\right)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d095a39392ba3ec9007fb5a583e75b253f10f718" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.021ex; height:6.176ex;" alt="{\displaystyle {\sqrt {\frac {2}{\pi }}}2^{\frac {z}{2}}\Gamma \left({\tfrac {z}{2}}+1\right)}"> </noscript><span class="lazy-image-placeholder" style="width: 17.021ex;height: 6.176ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d095a39392ba3ec9007fb5a583e75b253f10f718" data-alt="{\displaystyle {\sqrt {\frac {2}{\pi }}}2^{\frac {z}{2}}\Gamma \left({\tfrac {z}{2}}+1\right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  does not agree with the previous product formula for z!! for non-negative even integer values of z. Instead, this generalized formula implies the following alternative: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2k)!!={\sqrt {\frac {2}{\pi }}}2^{k}\Gamma \left(k+1\right)={\sqrt {\frac {2}{\pi }}}\prod _{i=1}^{k}(2i)\,,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> </msqrt> </mrow> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mi mathvariant="normal"> Γ </mi> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> </msqrt> </mrow> <munderover> <mo> ∏ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> i </mi> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (2k)!!={\sqrt {\frac {2}{\pi }}}2^{k}\Gamma \left(k+1\right)={\sqrt {\frac {2}{\pi }}}\prod _{i=1}^{k}(2i)\,,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4c29739a9346b0f2ee9c398cf6a68bf6cd1a0d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.936ex; height:7.343ex;" alt="{\displaystyle (2k)!!={\sqrt {\frac {2}{\pi }}}2^{k}\Gamma \left(k+1\right)={\sqrt {\frac {2}{\pi }}}\prod _{i=1}^{k}(2i)\,,}"> </noscript><span class="lazy-image-placeholder" style="width: 39.936ex;height: 7.343ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4c29739a9346b0f2ee9c398cf6a68bf6cd1a0d" data-alt="{\displaystyle (2k)!!={\sqrt {\frac {2}{\pi }}}2^{k}\Gamma \left(k+1\right)={\sqrt {\frac {2}{\pi }}}\prod _{i=1}^{k}(2i)\,,}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  with the value for 0!! in this case being <style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="plainlist" style="margin-left: 1.6em;"> <ul> <li> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0!!={\sqrt {\frac {2}{\pi }}}\approx 0.797\,884\,5608\dots }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 0 </mn> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> </msqrt> </mrow> <mo> ≈ </mo> <mn> 0.797 </mn> <mspace width="thinmathspace"></mspace> <mn> 884 </mn> <mspace width="thinmathspace"></mspace> <mn> 5608 </mn> <mo> … </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 0!!={\sqrt {\frac {2}{\pi }}}\approx 0.797\,884\,5608\dots } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d84c76e416688da30de4a0b7ffac578b0f5195aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.463ex; height:6.176ex;" alt="{\displaystyle 0!!={\sqrt {\frac {2}{\pi }}}\approx 0.797\,884\,5608\dots }"> </noscript><span class="lazy-image-placeholder" style="width: 30.463ex;height: 6.176ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d84c76e416688da30de4a0b7ffac578b0f5195aa" data-alt="{\displaystyle 0!!={\sqrt {\frac {2}{\pi }}}\approx 0.797\,884\,5608\dots }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (sequence <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://oeis.org/A076668" class="extiw" title="oeis:A076668">A076668</a> in the <a href="https://en-m-wikipedia-org.translate.goog/wiki/On-Line_Encyclopedia_of_Integer_Sequences?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</li> </ul> </div> Using this generalized formula as the definition, the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Volume_of_an_n-ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Volume of an n-ball">volume</a> of an n-<a href="https://en-m-wikipedia-org.translate.goog/wiki/Dimension?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dimension">dimensional</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hypersphere?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Hypersphere">hypersphere</a> of radius R can be expressed as<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-20">[20]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{n}={\frac {2\left(2\pi \right)^{\frac {n-1}{2}}}{n!!}}R^{n}\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> </mrow> </mfrac> </mrow> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V_{n}={\frac {2\left(2\pi \right)^{\frac {n-1}{2}}}{n!!}}R^{n}\,.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be406658f8eb6f3fa0835144706546676f48e00" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.565ex; height:7.176ex;" alt="{\displaystyle V_{n}={\frac {2\left(2\pi \right)^{\frac {n-1}{2}}}{n!!}}R^{n}\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 19.565ex;height: 7.176ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be406658f8eb6f3fa0835144706546676f48e00" data-alt="{\displaystyle V_{n}={\frac {2\left(2\pi \right)^{\frac {n-1}{2}}}{n!!}}R^{n}\,.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <h2 id="Additional_identities">Additional identities</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Double_factorial&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Additional identities" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> For integer values of n, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\frac {(n-1)!!}{n!!}}\times {\begin{cases}1&{\text{if }}n{\text{ is odd}}\\{\frac {\pi }{2}}&{\text{if }}n{\text{ is even.}}\end{cases}}}"><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"> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </msubsup> <msup> <mi> sin </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo> ⁡ </mo> <mi> x </mi> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> x </mi> <mo> = </mo> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </msubsup> <msup> <mi> cos </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo> ⁡ </mo> <mi> x </mi> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> x </mi> <mo> = </mo> <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> <mo> ! </mo> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> { </mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn> 1 </mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> if  </mtext> </mrow> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>  is odd </mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> if  </mtext> </mrow> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>  is even. </mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\frac {(n-1)!!}{n!!}}\times {\begin{cases}1&{\text{if }}n{\text{ is odd}}\\{\frac {\pi }{2}}&{\text{if }}n{\text{ is even.}}\end{cases}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a866cca59d475a6a2b67b12c36739d8b12ff2a3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:63.33ex; height:6.676ex;" alt="{\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\frac {(n-1)!!}{n!!}}\times {\begin{cases}1&{\text{if }}n{\text{ is odd}}\\{\frac {\pi }{2}}&{\text{if }}n{\text{ is even.}}\end{cases}}}"> </noscript><span class="lazy-image-placeholder" style="width: 63.33ex;height: 6.676ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a866cca59d475a6a2b67b12c36739d8b12ff2a3" data-alt="{\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\frac {(n-1)!!}{n!!}}\times {\begin{cases}1&{\text{if }}n{\text{ is odd}}\\{\frac {\pi }{2}}&{\text{if }}n{\text{ is even.}}\end{cases}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Using instead the extension of the double factorial of odd numbers to complex numbers, the formula is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\frac {(n-1)!!}{n!!}}{\sqrt {\frac {\pi }{2}}}\,.}"><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"> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </msubsup> <msup> <mi> sin </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo> ⁡ </mo> <mi> x </mi> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> x </mi> <mo> = </mo> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </msubsup> <msup> <mi> cos </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo> ⁡ </mo> <mi> x </mi> <mspace width="thinmathspace"></mspace> <mi> d </mi> <mi> x </mi> <mo> = </mo> <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> <mo> ! </mo> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\frac {(n-1)!!}{n!!}}{\sqrt {\frac {\pi }{2}}}\,.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf50e6c2f2af94ef55fede8192f95820b873a2d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.131ex; height:6.843ex;" alt="{\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\frac {(n-1)!!}{n!!}}{\sqrt {\frac {\pi }{2}}}\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 48.131ex;height: 6.843ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf50e6c2f2af94ef55fede8192f95820b873a2d8" data-alt="{\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\frac {(n-1)!!}{n!!}}{\sqrt {\frac {\pi }{2}}}\,.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Double factorials can also be used to evaluate integrals of more complicated trigonometric polynomials.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-meserve-9">[9]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-21">[21]</a> Double factorials of odd numbers are related to the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gamma_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gamma function">gamma function</a> by the identity: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2n-1)!!=2^{n}\cdot {\frac {\Gamma \left({\frac {1}{2}}+n\right)}{\sqrt {\pi }}}=(-2)^{n}\cdot {\frac {\sqrt {\pi }}{\Gamma \left({\frac {1}{2}}-n\right)}}\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> Γ </mi> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mi> π </mi> </msqrt> </mfrac> </mrow> <mo> = </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 2 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi> π </mi> </msqrt> <mrow> <mi mathvariant="normal"> Γ </mi> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> − </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (2n-1)!!=2^{n}\cdot {\frac {\Gamma \left({\frac {1}{2}}+n\right)}{\sqrt {\pi }}}=(-2)^{n}\cdot {\frac {\sqrt {\pi }}{\Gamma \left({\frac {1}{2}}-n\right)}}\,.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1010dd65bb58c9f4c0fe3d2a7115916e523fa77" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:50.029ex; height:7.843ex;" alt="{\displaystyle (2n-1)!!=2^{n}\cdot {\frac {\Gamma \left({\frac {1}{2}}+n\right)}{\sqrt {\pi }}}=(-2)^{n}\cdot {\frac {\sqrt {\pi }}{\Gamma \left({\frac {1}{2}}-n\right)}}\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 50.029ex;height: 7.843ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1010dd65bb58c9f4c0fe3d2a7115916e523fa77" data-alt="{\displaystyle (2n-1)!!=2^{n}\cdot {\frac {\Gamma \left({\frac {1}{2}}+n\right)}{\sqrt {\pi }}}=(-2)^{n}\cdot {\frac {\sqrt {\pi }}{\Gamma \left({\frac {1}{2}}-n\right)}}\,.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Some additional identities involving double factorials of odd numbers are:<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-callan-1">[1]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(2n-1)!!&=\sum _{k=0}^{n-1}{\binom {n}{k+1}}(2k-1)!!(2n-2k-3)!!\\&=\sum _{k=1}^{n}{\binom {n}{k}}(2k-3)!!(2(n-k)-1)!!\\&=\sum _{k=0}^{n}{\binom {2n-k-1}{k-1}}{\frac {(2k-1)(2n-k+1)}{k+1}}(2n-2k-3)!!\\&=\sum _{k=1}^{n}{\frac {(n-1)!}{(k-1)!}}k(2k-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> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <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> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> n </mi> <mo> − </mo> <mn> 2 </mn> <mi> k </mi> <mo> − </mo> <mn> 3 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo> − </mo> <mn> 3 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mi> k </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </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> <mn> 2 </mn> <mi> n </mi> <mo> − </mo> <mi> k </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"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> n </mi> <mo> − </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> n </mi> <mo> − </mo> <mn> 2 </mn> <mi> k </mi> <mo> − </mo> <mn> 3 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> </mrow> <mrow> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> </mrow> </mfrac> </mrow> <mi> k </mi> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo> − </mo> <mn> 3 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}(2n-1)!!&=\sum _{k=0}^{n-1}{\binom {n}{k+1}}(2k-1)!!(2n-2k-3)!!\\&=\sum _{k=1}^{n}{\binom {n}{k}}(2k-3)!!(2(n-k)-1)!!\\&=\sum _{k=0}^{n}{\binom {2n-k-1}{k-1}}{\frac {(2k-1)(2n-k+1)}{k+1}}(2n-2k-3)!!\\&=\sum _{k=1}^{n}{\frac {(n-1)!}{(k-1)!}}k(2k-3)!!\,.\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7433063a81181baaf577f96dab8938f0359f09d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.838ex; width:67.608ex; height:28.843ex;" alt="{\displaystyle {\begin{aligned}(2n-1)!!&=\sum _{k=0}^{n-1}{\binom {n}{k+1}}(2k-1)!!(2n-2k-3)!!\\&=\sum _{k=1}^{n}{\binom {n}{k}}(2k-3)!!(2(n-k)-1)!!\\&=\sum _{k=0}^{n}{\binom {2n-k-1}{k-1}}{\frac {(2k-1)(2n-k+1)}{k+1}}(2n-2k-3)!!\\&=\sum _{k=1}^{n}{\frac {(n-1)!}{(k-1)!}}k(2k-3)!!\,.\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 67.608ex;height: 28.843ex;vertical-align: -13.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7433063a81181baaf577f96dab8938f0359f09d" data-alt="{\displaystyle {\begin{aligned}(2n-1)!!&=\sum _{k=0}^{n-1}{\binom {n}{k+1}}(2k-1)!!(2n-2k-3)!!\\&=\sum _{k=1}^{n}{\binom {n}{k}}(2k-3)!!(2(n-k)-1)!!\\&=\sum _{k=0}^{n}{\binom {2n-k-1}{k-1}}{\frac {(2k-1)(2n-k+1)}{k+1}}(2n-2k-3)!!\\&=\sum _{k=1}^{n}{\frac {(n-1)!}{(k-1)!}}k(2k-3)!!\,.\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  An approximation for the ratio of the double factorial of two consecutive integers is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(2n)!!}{(2n-1)!!}}\approx {\sqrt {\pi n}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> n </mi> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> </mrow> <mrow> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> ≈ </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi> π </mi> <mi> n </mi> </msqrt> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {(2n)!!}{(2n-1)!!}}\approx {\sqrt {\pi n}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181ad38fcf41b55fc6202b1460030ab1817b33b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.907ex; height:6.509ex;" alt="{\displaystyle {\frac {(2n)!!}{(2n-1)!!}}\approx {\sqrt {\pi n}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 18.907ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181ad38fcf41b55fc6202b1460030ab1817b33b4" data-alt="{\displaystyle {\frac {(2n)!!}{(2n-1)!!}}\approx {\sqrt {\pi n}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  This approximation gets more accurate as n increases, which can be seen as a result of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wallis%27_integrals?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Deducing_the_Double_Factorial_Ratio" title="Wallis' integrals"> Wallis Integral</a>. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"> <h2 id="Generalizations">Generalizations</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Double_factorial&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Generalizations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-7 collapsible-block" id="mf-section-7"> <div class="mw-heading mw-heading3"> <h3 id="Definitions">Definitions</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Double_factorial&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Definitions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> In the same way that the double factorial generalizes the notion of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Factorial">single factorial</a>, the following definition of the integer-valued multiple factorial functions (multifactorials), or α-factorial functions, extends the notion of the double factorial function for positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> α </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \alpha } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> </noscript><span class="lazy-image-placeholder" style="width: 1.488ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" data-alt="{\displaystyle \alpha }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> : <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!_{(\alpha )}={\begin{cases}n\cdot (n-\alpha )!_{(\alpha )}&{\text{ if }}n>\alpha \,;\\n&{\text{ if }}1\leq n\leq \alpha \,;{\text{and}}\\(n+\alpha )!_{(\alpha )}/(n+\alpha )&{\text{ if }}n\leq 0{\text{ and is not a negative multiple of }}\alpha \,;\end{cases}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <msub> <mo> ! </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> { </mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi> n </mi> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> <msub> <mo> ! </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>  if  </mtext> </mrow> <mi> n </mi> <mo> > </mo> <mi> α </mi> <mspace width="thinmathspace"></mspace> <mo> ; </mo> </mtd> </mtr> <mtr> <mtd> <mi> n </mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>  if  </mtext> </mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> n </mi> <mo> ≤ </mo> <mi> α </mi> <mspace width="thinmathspace"></mspace> <mo> ; </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> + </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> <msub> <mo> ! </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> + </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>  if  </mtext> </mrow> <mi> n </mi> <mo> ≤ </mo> <mn> 0 </mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>  and is not a negative multiple of  </mtext> </mrow> <mi> α </mi> <mspace width="thinmathspace"></mspace> <mo> ; </mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!_{(\alpha )}={\begin{cases}n\cdot (n-\alpha )!_{(\alpha )}&{\text{ if }}n>\alpha \,;\\n&{\text{ if }}1\leq n\leq \alpha \,;{\text{and}}\\(n+\alpha )!_{(\alpha )}/(n+\alpha )&{\text{ if }}n\leq 0{\text{ and is not a negative multiple of }}\alpha \,;\end{cases}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f38a6d35bb70e2ec2e5b7996a3298779ea7ee2e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.749ex; margin-bottom: -0.256ex; width:76.189ex; height:9.176ex;" alt="{\displaystyle n!_{(\alpha )}={\begin{cases}n\cdot (n-\alpha )!_{(\alpha )}&{\text{ if }}n>\alpha \,;\\n&{\text{ if }}1\leq n\leq \alpha \,;{\text{and}}\\(n+\alpha )!_{(\alpha )}/(n+\alpha )&{\text{ if }}n\leq 0{\text{ and is not a negative multiple of }}\alpha \,;\end{cases}}}"> </noscript><span class="lazy-image-placeholder" style="width: 76.189ex;height: 9.176ex;vertical-align: -3.749ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f38a6d35bb70e2ec2e5b7996a3298779ea7ee2e9" data-alt="{\displaystyle n!_{(\alpha )}={\begin{cases}n\cdot (n-\alpha )!_{(\alpha )}&{\text{ if }}n>\alpha \,;\\n&{\text{ if }}1\leq n\leq \alpha \,;{\text{and}}\\(n+\alpha )!_{(\alpha )}/(n+\alpha )&{\text{ if }}n\leq 0{\text{ and is not a negative multiple of }}\alpha \,;\end{cases}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  <div class="mw-heading mw-heading3"> <h3 id="Alternative_extension_of_the_multifactorial">Alternative extension of the multifactorial</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Double_factorial&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Alternative extension of the multifactorial" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Alternatively, the multifactorial z!(α) can be extended to most real and complex numbers z by noting that when z is one more than a positive multiple of the positive integer α then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}z!_{(\alpha )}&=z(z-\alpha )\cdots (\alpha +1)\\&=\alpha ^{\frac {z-1}{\alpha }}\left({\frac {z}{\alpha }}\right)\left({\frac {z-\alpha }{\alpha }}\right)\cdots \left({\frac {\alpha +1}{\alpha }}\right)\\&=\alpha ^{\frac {z-1}{\alpha }}{\frac {\Gamma \left({\frac {z}{\alpha }}+1\right)}{\Gamma \left({\frac {1}{\alpha }}+1\right)}}\,.\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> z </mi> <msub> <mo> ! </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo> − </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> <mo> ⋯ </mo> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <msup> <mi> α </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> z </mi> <mo> − </mo> <mn> 1 </mn> </mrow> <mi> α </mi> </mfrac> </mrow> </msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> z </mi> <mi> α </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> z </mi> <mo> − </mo> <mi> α </mi> </mrow> <mi> α </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⋯ </mo> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> α </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <msup> <mi> α </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> z </mi> <mo> − </mo> <mn> 1 </mn> </mrow> <mi> α </mi> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> Γ </mi> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> z </mi> <mi> α </mi> </mfrac> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi mathvariant="normal"> Γ </mi> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> α </mi> </mfrac> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}z!_{(\alpha )}&=z(z-\alpha )\cdots (\alpha +1)\\&=\alpha ^{\frac {z-1}{\alpha }}\left({\frac {z}{\alpha }}\right)\left({\frac {z-\alpha }{\alpha }}\right)\cdots \left({\frac {\alpha +1}{\alpha }}\right)\\&=\alpha ^{\frac {z-1}{\alpha }}{\frac {\Gamma \left({\frac {z}{\alpha }}+1\right)}{\Gamma \left({\frac {1}{\alpha }}+1\right)}}\,.\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56dc326b0c8f1265bd560f78e541886da1e0ce46" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:41.827ex; height:17.176ex;" alt="{\displaystyle {\begin{aligned}z!_{(\alpha )}&=z(z-\alpha )\cdots (\alpha +1)\\&=\alpha ^{\frac {z-1}{\alpha }}\left({\frac {z}{\alpha }}\right)\left({\frac {z-\alpha }{\alpha }}\right)\cdots \left({\frac {\alpha +1}{\alpha }}\right)\\&=\alpha ^{\frac {z-1}{\alpha }}{\frac {\Gamma \left({\frac {z}{\alpha }}+1\right)}{\Gamma \left({\frac {1}{\alpha }}+1\right)}}\,.\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 41.827ex;height: 17.176ex;vertical-align: -8.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56dc326b0c8f1265bd560f78e541886da1e0ce46" data-alt="{\displaystyle {\begin{aligned}z!_{(\alpha )}&=z(z-\alpha )\cdots (\alpha +1)\\&=\alpha ^{\frac {z-1}{\alpha }}\left({\frac {z}{\alpha }}\right)\left({\frac {z-\alpha }{\alpha }}\right)\cdots \left({\frac {\alpha +1}{\alpha }}\right)\\&=\alpha ^{\frac {z-1}{\alpha }}{\frac {\Gamma \left({\frac {z}{\alpha }}+1\right)}{\Gamma \left({\frac {1}{\alpha }}+1\right)}}\,.\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  This last expression is defined much more broadly than the original. In the same way that z! is not defined for negative integers, and z‼ is not defined for negative even integers, z!(α) is not defined for negative multiples of α. However, it is defined and satisfies (z+α)!(α) = (z+α)·z!(α) for all other complex numbers z. This definition is consistent with the earlier definition only for those integers z satisfying z ≡ 1 mod α. In addition to extending z!(α) to most complex numbers z, this definition has the feature of working for all positive real values of α. Furthermore, when α = 1, this definition is mathematically equivalent to the Π(z) function, described above. Also, when α = 2, this definition is mathematically equivalent to the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Complex_arguments">alternative extension of the double factorial</a>. <div class="mw-heading mw-heading3"> <h3 id="Generalized_Stirling_numbers_expanding_the_multifactorial_functions">Generalized Stirling numbers expanding the multifactorial functions</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Double_factorial&action=edit&section=12&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Generalized Stirling numbers expanding the multifactorial functions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> A class of generalized <a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling_numbers_of_the_first_kind?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Stirling numbers of the first kind">Stirling numbers of the first kind</a> is defined for α > 0 by the following triangular recurrence relation: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]_{\alpha }=(\alpha n+1-2\alpha )\left[{\begin{matrix}n-1\\k\end{matrix}}\right]_{\alpha }+\left[{\begin{matrix}n-1\\k-1\end{matrix}}\right]_{\alpha }+\delta _{n,0}\delta _{k,0}\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> α </mi> </mrow> </msub> <mo> = </mo> <mo stretchy="false"> ( </mo> <mi> α </mi> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> <mo> − </mo> <mn> 2 </mn> <mi> α </mi> <mo stretchy="false"> ) </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> </mtd> </mtr> </mtable> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> α </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> α </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> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]_{\alpha }=(\alpha n+1-2\alpha )\left[{\begin{matrix}n-1\\k\end{matrix}}\right]_{\alpha }+\left[{\begin{matrix}n-1\\k-1\end{matrix}}\right]_{\alpha }+\delta _{n,0}\delta _{k,0}\,.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/266e3cc9a98d1308cb1e10fcdeb31bf748882f6f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.591ex; height:6.176ex;" alt="{\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]_{\alpha }=(\alpha n+1-2\alpha )\left[{\begin{matrix}n-1\\k\end{matrix}}\right]_{\alpha }+\left[{\begin{matrix}n-1\\k-1\end{matrix}}\right]_{\alpha }+\delta _{n,0}\delta _{k,0}\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 56.591ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/266e3cc9a98d1308cb1e10fcdeb31bf748882f6f" data-alt="{\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]_{\alpha }=(\alpha n+1-2\alpha )\left[{\begin{matrix}n-1\\k\end{matrix}}\right]_{\alpha }+\left[{\begin{matrix}n-1\\k-1\end{matrix}}\right]_{\alpha }+\delta _{n,0}\delta _{k,0}\,.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  These generalized α-factorial coefficients then generate the distinct symbolic polynomial products defining the multiple factorial, or α-factorial functions, (x − 1)!(α), as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(x-1|\alpha )^{\underline {n}}&:=\prod _{i=0}^{n-1}\left(x-1-i\alpha \right)\\&=(x-1)(x-1-\alpha )\cdots {\bigl (}x-1-(n-1)\alpha {\bigr )}\\&=\sum _{k=0}^{n}\left[{\begin{matrix}n\\k\end{matrix}}\right](-\alpha )^{n-k}(x-1)^{k}\\&=\sum _{k=1}^{n}\left[{\begin{matrix}n\\k\end{matrix}}\right]_{\alpha }(-1)^{n-k}x^{k-1}\,.\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> − </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> α </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi> n </mi> <mo> _ </mo> </munder> </mrow> </msup> </mtd> <mtd> <mi></mi> <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> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> − </mo> <mn> 1 </mn> <mo> − </mo> <mi> i </mi> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </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> 1 </mn> <mo> − </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> <mo> ⋯ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mi> x </mi> <mo> − </mo> <mn> 1 </mn> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mi> α </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <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 stretchy="false"> ( </mo> <mo> − </mo> <mi> α </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> − </mo> <mi> k </mi> </mrow> </msup> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> − </mo> <mn> 1 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <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> α </mi> </mrow> </msub> <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> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}(x-1|\alpha )^{\underline {n}}&:=\prod _{i=0}^{n-1}\left(x-1-i\alpha \right)\\&=(x-1)(x-1-\alpha )\cdots {\bigl (}x-1-(n-1)\alpha {\bigr )}\\&=\sum _{k=0}^{n}\left[{\begin{matrix}n\\k\end{matrix}}\right](-\alpha )^{n-k}(x-1)^{k}\\&=\sum _{k=1}^{n}\left[{\begin{matrix}n\\k\end{matrix}}\right]_{\alpha }(-1)^{n-k}x^{k-1}\,.\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abea228f6758116d56a58d09d51927b45a7bd1ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.731ex; margin-bottom: -0.273ex; width:55.453ex; height:25.176ex;" alt="{\displaystyle {\begin{aligned}(x-1|\alpha )^{\underline {n}}&:=\prod _{i=0}^{n-1}\left(x-1-i\alpha \right)\\&=(x-1)(x-1-\alpha )\cdots {\bigl (}x-1-(n-1)\alpha {\bigr )}\\&=\sum _{k=0}^{n}\left[{\begin{matrix}n\\k\end{matrix}}\right](-\alpha )^{n-k}(x-1)^{k}\\&=\sum _{k=1}^{n}\left[{\begin{matrix}n\\k\end{matrix}}\right]_{\alpha }(-1)^{n-k}x^{k-1}\,.\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 55.453ex;height: 25.176ex;vertical-align: -11.731ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abea228f6758116d56a58d09d51927b45a7bd1ff" data-alt="{\displaystyle {\begin{aligned}(x-1|\alpha )^{\underline {n}}&:=\prod _{i=0}^{n-1}\left(x-1-i\alpha \right)\\&=(x-1)(x-1-\alpha )\cdots {\bigl (}x-1-(n-1)\alpha {\bigr )}\\&=\sum _{k=0}^{n}\left[{\begin{matrix}n\\k\end{matrix}}\right](-\alpha )^{n-k}(x-1)^{k}\\&=\sum _{k=1}^{n}\left[{\begin{matrix}n\\k\end{matrix}}\right]_{\alpha }(-1)^{n-k}x^{k-1}\,.\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The distinct polynomial expansions in the previous equations actually define the α-factorial products for multiple distinct cases of the least residues x ≡ n0 mod α for n0 ∈ {0, 1, 2, ..., α − 1}. The generalized α-factorial polynomials, σ(α) n(x) where σ(1) n(x) ≡ σn(x), which generalize the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Stirling_polynomial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Stirling_convolution_polynomials" class="mw-redirect" title="Stirling polynomial">Stirling convolution polynomials</a> from the single factorial case to the multifactorial cases, are defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{n}^{(\alpha )}(x):=\left[{\begin{matrix}x\\x-n\end{matrix}}\right]_{(\alpha )}{\frac {(x-n-1)!}{x!}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> := </mo> <msub> <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> <mi> n </mi> </mtd> </mtr> </mtable> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> − </mo> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> </mrow> <mrow> <mi> x </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sigma _{n}^{(\alpha )}(x):=\left[{\begin{matrix}x\\x-n\end{matrix}}\right]_{(\alpha )}{\frac {(x-n-1)!}{x!}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea63311a69cdf64bb4848a9fa5a44ce2b30d5d17" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.973ex; height:6.843ex;" alt="{\displaystyle \sigma _{n}^{(\alpha )}(x):=\left[{\begin{matrix}x\\x-n\end{matrix}}\right]_{(\alpha )}{\frac {(x-n-1)!}{x!}}}"> </noscript><span class="lazy-image-placeholder" style="width: 34.973ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea63311a69cdf64bb4848a9fa5a44ce2b30d5d17" data-alt="{\displaystyle \sigma _{n}^{(\alpha )}(x):=\left[{\begin{matrix}x\\x-n\end{matrix}}\right]_{(\alpha )}{\frac {(x-n-1)!}{x!}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  for 0 ≤ n ≤ x. These polynomials have a particularly nice closed-form <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ordinary_generating_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Ordinary generating function">ordinary generating function</a> given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n\geq 0}x\cdot \sigma _{n}^{(\alpha )}(x)z^{n}=e^{(1-\alpha )z}\left({\frac {\alpha ze^{\alpha z}}{e^{\alpha z}-1}}\right)^{x}\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> ≥ </mo> <mn> 0 </mn> </mrow> </munder> <mi> x </mi> <mo> ⋅ </mo> <msubsup> <mi> σ </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo> = </mo> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> − </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> <mi> z </mi> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> α </mi> <mi> z </mi> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> α </mi> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <msup> <mi> e </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> α </mi> <mi> z </mi> </mrow> </msup> <mo> − </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sum _{n\geq 0}x\cdot \sigma _{n}^{(\alpha )}(x)z^{n}=e^{(1-\alpha )z}\left({\frac {\alpha ze^{\alpha z}}{e^{\alpha z}-1}}\right)^{x}\,.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/558323af82ab55031d091b7bcfc5fc661e66ea0c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.312ex; height:6.843ex;" alt="{\displaystyle \sum _{n\geq 0}x\cdot \sigma _{n}^{(\alpha )}(x)z^{n}=e^{(1-\alpha )z}\left({\frac {\alpha ze^{\alpha z}}{e^{\alpha z}-1}}\right)^{x}\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 39.312ex;height: 6.843ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/558323af82ab55031d091b7bcfc5fc661e66ea0c" data-alt="{\displaystyle \sum _{n\geq 0}x\cdot \sigma _{n}^{(\alpha )}(x)z^{n}=e^{(1-\alpha )z}\left({\frac {\alpha ze^{\alpha z}}{e^{\alpha z}-1}}\right)^{x}\,.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Other combinatorial properties and expansions of these generalized α-factorial triangles and polynomial sequences are considered in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFSchmidt2010">Schmidt (2010)</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-22">[22]</a> <div class="mw-heading mw-heading3"> <h3 id="Exact_finite_sums_involving_multiple_factorial_functions">Exact finite sums involving multiple factorial functions</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Double_factorial&action=edit&section=13&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Exact finite sums involving multiple factorial functions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Suppose that n ≥ 1 and α ≥ 2 are integer-valued. Then we can expand the next single finite sums involving the multifactorial, or α-factorial functions, (αn − 1)!(α), in terms of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Pochhammer_symbol?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Pochhammer symbol">Pochhammer symbol</a> and the generalized, rational-valued <a href="https://en-m-wikipedia-org.translate.goog/wiki/Binomial_coefficients?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Binomial coefficients">binomial coefficients</a> as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(\alpha n-1)!_{(\alpha )}&=\sum _{k=0}^{n-1}{\binom {n-1}{k+1}}(-1)^{k}\times \left({\frac {1}{\alpha }}\right)_{-(k+1)}\left({\frac {1}{\alpha }}-n\right)_{k+1}\times {\bigl (}\alpha (k+1)-1{\bigr )}!_{(\alpha )}{\bigl (}\alpha (n-k-1)-1{\bigr )}!_{(\alpha )}\\&=\sum _{k=0}^{n-1}{\binom {n-1}{k+1}}(-1)^{k}\times {\binom {{\frac {1}{\alpha }}+k-n}{k+1}}{\binom {{\frac {1}{\alpha }}-1}{k+1}}\times {\bigl (}\alpha (k+1)-1{\bigr )}!_{(\alpha )}{\bigl (}\alpha (n-k-1)-1{\bigr )}!_{(\alpha )}\,,\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> <mo stretchy="false"> ( </mo> <mi> α </mi> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <msub> <mo> ! </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mrow> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> <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> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 1 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo> × </mo> <msub> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> α </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msub> <msub> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> α </mi> </mfrac> </mrow> <mo> − </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mi> α </mi> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> − </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> <msub> <mo> ! </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mi> α </mi> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> − </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> <msub> <mo> ! </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msub> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mrow> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> <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> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 1 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> α </mi> </mfrac> </mrow> <mo> + </mo> <mi> k </mi> <mo> − </mo> <mi> n </mi> </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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mi> α </mi> </mfrac> </mrow> <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> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mi> α </mi> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> − </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> <msub> <mo> ! </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mi> α </mi> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> − </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> <msub> <mo> ! </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mo> , </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}(\alpha n-1)!_{(\alpha )}&=\sum _{k=0}^{n-1}{\binom {n-1}{k+1}}(-1)^{k}\times \left({\frac {1}{\alpha }}\right)_{-(k+1)}\left({\frac {1}{\alpha }}-n\right)_{k+1}\times {\bigl (}\alpha (k+1)-1{\bigr )}!_{(\alpha )}{\bigl (}\alpha (n-k-1)-1{\bigr )}!_{(\alpha )}\\&=\sum _{k=0}^{n-1}{\binom {n-1}{k+1}}(-1)^{k}\times {\binom {{\frac {1}{\alpha }}+k-n}{k+1}}{\binom {{\frac {1}{\alpha }}-1}{k+1}}\times {\bigl (}\alpha (k+1)-1{\bigr )}!_{(\alpha )}{\bigl (}\alpha (n-k-1)-1{\bigr )}!_{(\alpha )}\,,\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc4cb1c887065c7681d2d4e769bc46a196927c5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:104.47ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}(\alpha n-1)!_{(\alpha )}&=\sum _{k=0}^{n-1}{\binom {n-1}{k+1}}(-1)^{k}\times \left({\frac {1}{\alpha }}\right)_{-(k+1)}\left({\frac {1}{\alpha }}-n\right)_{k+1}\times {\bigl (}\alpha (k+1)-1{\bigr )}!_{(\alpha )}{\bigl (}\alpha (n-k-1)-1{\bigr )}!_{(\alpha )}\\&=\sum _{k=0}^{n-1}{\binom {n-1}{k+1}}(-1)^{k}\times {\binom {{\frac {1}{\alpha }}+k-n}{k+1}}{\binom {{\frac {1}{\alpha }}-1}{k+1}}\times {\bigl (}\alpha (k+1)-1{\bigr )}!_{(\alpha )}{\bigl (}\alpha (n-k-1)-1{\bigr )}!_{(\alpha )}\,,\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 104.47ex;height: 14.843ex;vertical-align: -6.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc4cb1c887065c7681d2d4e769bc46a196927c5" data-alt="{\displaystyle {\begin{aligned}(\alpha n-1)!_{(\alpha )}&=\sum _{k=0}^{n-1}{\binom {n-1}{k+1}}(-1)^{k}\times \left({\frac {1}{\alpha }}\right)_{-(k+1)}\left({\frac {1}{\alpha }}-n\right)_{k+1}\times {\bigl (}\alpha (k+1)-1{\bigr )}!_{(\alpha )}{\bigl (}\alpha (n-k-1)-1{\bigr )}!_{(\alpha )}\\&=\sum _{k=0}^{n-1}{\binom {n-1}{k+1}}(-1)^{k}\times {\binom {{\frac {1}{\alpha }}+k-n}{k+1}}{\binom {{\frac {1}{\alpha }}-1}{k+1}}\times {\bigl (}\alpha (k+1)-1{\bigr )}!_{(\alpha )}{\bigl (}\alpha (n-k-1)-1{\bigr )}!_{(\alpha )}\,,\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  and moreover, we similarly have double sum expansions of these functions given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(\alpha n-1)!_{(\alpha )}&=\sum _{k=0}^{n-1}\sum _{i=0}^{k+1}{\binom {n-1}{k+1}}{\binom {k+1}{i}}(-1)^{k}\alpha ^{k+1-i}(\alpha i-1)!_{(\alpha )}{\bigl (}\alpha (n-1-k)-1{\bigr )}!_{(\alpha )}\times (n-1-k)_{k+1-i}\\&=\sum _{k=0}^{n-1}\sum _{i=0}^{k+1}{\binom {n-1}{k+1}}{\binom {k+1}{i}}{\binom {n-1-i}{k+1-i}}(-1)^{k}\alpha ^{k+1-i}(\alpha i-1)!_{(\alpha )}{\bigl (}\alpha (n-1-k)-1{\bigr )}!_{(\alpha )}\times (k+1-i)!.\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> <mo stretchy="false"> ( </mo> <mi> α </mi> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <msub> <mo> ! </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </munderover> <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> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mrow> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> <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> <mi> k </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> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 1 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <msup> <mi> α </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> <mo> − </mo> <mi> i </mi> </mrow> </msup> <mo stretchy="false"> ( </mo> <mi> α </mi> <mi> i </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <msub> <mo> ! </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mi> α </mi> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo> − </mo> <mi> k </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> <msub> <mo> ! </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo> − </mo> <mi> k </mi> <msub> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> <mo> − </mo> <mi> i </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </munderover> <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> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mrow> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> <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> <mi> k </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"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mrow> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo> − </mo> <mi> i </mi> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> <mo> − </mo> <mi> i </mi> </mrow> </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> k </mi> </mrow> </msup> <msup> <mi> α </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> <mo> − </mo> <mi> i </mi> </mrow> </msup> <mo stretchy="false"> ( </mo> <mi> α </mi> <mi> i </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <msub> <mo> ! </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mi> α </mi> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo> − </mo> <mi> k </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> <msub> <mo> ! </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> <mo> − </mo> <mi> i </mi> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}(\alpha n-1)!_{(\alpha )}&=\sum _{k=0}^{n-1}\sum _{i=0}^{k+1}{\binom {n-1}{k+1}}{\binom {k+1}{i}}(-1)^{k}\alpha ^{k+1-i}(\alpha i-1)!_{(\alpha )}{\bigl (}\alpha (n-1-k)-1{\bigr )}!_{(\alpha )}\times (n-1-k)_{k+1-i}\\&=\sum _{k=0}^{n-1}\sum _{i=0}^{k+1}{\binom {n-1}{k+1}}{\binom {k+1}{i}}{\binom {n-1-i}{k+1-i}}(-1)^{k}\alpha ^{k+1-i}(\alpha i-1)!_{(\alpha )}{\bigl (}\alpha (n-1-k)-1{\bigr )}!_{(\alpha )}\times (k+1-i)!.\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0eec0e1968ed1890790fa38b1526ab6fad7d313" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:113.748ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}(\alpha n-1)!_{(\alpha )}&=\sum _{k=0}^{n-1}\sum _{i=0}^{k+1}{\binom {n-1}{k+1}}{\binom {k+1}{i}}(-1)^{k}\alpha ^{k+1-i}(\alpha i-1)!_{(\alpha )}{\bigl (}\alpha (n-1-k)-1{\bigr )}!_{(\alpha )}\times (n-1-k)_{k+1-i}\\&=\sum _{k=0}^{n-1}\sum _{i=0}^{k+1}{\binom {n-1}{k+1}}{\binom {k+1}{i}}{\binom {n-1-i}{k+1-i}}(-1)^{k}\alpha ^{k+1-i}(\alpha i-1)!_{(\alpha )}{\bigl (}\alpha (n-1-k)-1{\bigr )}!_{(\alpha )}\times (k+1-i)!.\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 113.748ex;height: 14.843ex;vertical-align: -6.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0eec0e1968ed1890790fa38b1526ab6fad7d313" data-alt="{\displaystyle {\begin{aligned}(\alpha n-1)!_{(\alpha )}&=\sum _{k=0}^{n-1}\sum _{i=0}^{k+1}{\binom {n-1}{k+1}}{\binom {k+1}{i}}(-1)^{k}\alpha ^{k+1-i}(\alpha i-1)!_{(\alpha )}{\bigl (}\alpha (n-1-k)-1{\bigr )}!_{(\alpha )}\times (n-1-k)_{k+1-i}\\&=\sum _{k=0}^{n-1}\sum _{i=0}^{k+1}{\binom {n-1}{k+1}}{\binom {k+1}{i}}{\binom {n-1-i}{k+1-i}}(-1)^{k}\alpha ^{k+1-i}(\alpha i-1)!_{(\alpha )}{\bigl (}\alpha (n-1-k)-1{\bigr )}!_{(\alpha )}\times (k+1-i)!.\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The first two sums above are similar in form to a known non-round combinatorial identity for the double factorial function when α := 2 given by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFCallan2009">Callan (2009)</a>. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2n-1)!!=\sum _{k=0}^{n-1}{\binom {n}{k+1}}(2k-1)!!(2n-2k-3)!!.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <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> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> n </mi> <mo> − </mo> <mn> 2 </mn> <mi> k </mi> <mo> − </mo> <mn> 3 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (2n-1)!!=\sum _{k=0}^{n-1}{\binom {n}{k+1}}(2k-1)!!(2n-2k-3)!!.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed043899ec33bde26c4ddaef9b9aff69f49229f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:50.142ex; height:7.509ex;" alt="{\displaystyle (2n-1)!!=\sum _{k=0}^{n-1}{\binom {n}{k+1}}(2k-1)!!(2n-2k-3)!!.}"> </noscript><span class="lazy-image-placeholder" style="width: 50.142ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed043899ec33bde26c4ddaef9b9aff69f49229f" data-alt="{\displaystyle (2n-1)!!=\sum _{k=0}^{n-1}{\binom {n}{k+1}}(2k-1)!!(2n-2k-3)!!.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Similar identities can be obtained via context-free grammars.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-23">[23]</a> Additional finite sum expansions of congruences for the α-factorial functions, (αn − d)!(α), modulo any prescribed integer h ≥ 2 for any 0 ≤ d < α are given by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFSchmidt2018">Schmidt (2018)</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-24">[24]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"> <h2 id="References">References</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Double_factorial&action=edit&section=14&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-8 collapsible-block" id="mf-section-8"> <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-callan-1">^ <a 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href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-callan_1-7">h</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-callan_1-8">i</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-callan_1-9">j</a> <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="CITEREFCallan2009" class="citation arxiv cs1">Callan, David (2009). 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Proceedings of the Royal Society of London. 71 (467–476): 97–101. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1098%252Frspl.1902.0068">10.1098/rspl.1902.0068</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/JSTOR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.jstor.org/stable/116355">116355</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+London&rft.atitle=On+some+definite+integrals+and+a+new+method+of+reducing+a+function+of+spherical+co-ordinates+to+a+series+of+spherical+harmonics&rft.volume=71&rft.issue=467%E2%80%93476&rft.pages=97-101&rft.date=1902&rft_id=info%3Adoi%2F10.1098%2Frspl.1902.0068&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F116355%23id-name%3DJSTOR&rft.aulast=Schuster&rft.aufirst=Arthur&rft_id=https%3A%2F%2Fdoi.org%2F10.1098%252Frspl.1902.0068&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADouble+factorial" class="Z3988"> See in particular p. 99.</li> <li id="cite_note-meserve-9">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-meserve_9-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-meserve_9-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeserve1948" class="citation journal cs1">Meserve, B. 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"Tiling bijections between paths and Brauer diagrams". <a href="https://en-m-wikipedia-org.translate.goog/wiki/Journal_of_Algebraic_Combinatorics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Journal of Algebraic Combinatorics">Journal of Algebraic Combinatorics</a>. 33 (3): 427–453. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ArXiv_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://arxiv.org/abs/0906.0912">0906.0912</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007%252Fs10801-010-0252-6">10.1007/s10801-010-0252-6</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/MR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathscinet.ams.org/mathscinet-getitem?mr%3D2772541">2772541</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/S2CID_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://api.semanticscholar.org/CorpusID:7264692">7264692</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+Algebraic+Combinatorics&rft.atitle=Tiling+bijections+between+paths+and+Brauer+diagrams&rft.volume=33&rft.issue=3&rft.pages=427-453&rft.date=2011&rft_id=info%3Aarxiv%2F0906.0912&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2772541%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7264692%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs10801-010-0252-6&rft.aulast=Marsh&rft.aufirst=Robert+J.&rft.au=Martin%2C+Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADouble+factorial" class="Z3988"></li> <li id="cite_note-18"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHassani2000" class="citation book cs1">Hassani, Sadri (2000). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DdxSOzeLMij4C%26pg%3DPA266">Mathematical Methods: For Students of Physics and Related Fields</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Undergraduate_Texts_in_Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a>. 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Bulletin de la Société Mathématique de Grèce. 28 (A): 40–43. <a href="https://en-m-wikipedia-org.translate.goog/wiki/MR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathscinet.ams.org/mathscinet-getitem?mr%3D0935868">0935868</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+de+la+Soci%C3%A9t%C3%A9+Math%C3%A9matique+de+Gr%C3%A8ce&rft.atitle=A+useful+application+of+Gauss+theorem&rft.volume=28&rft.issue=A&rft.pages=40-43&rft.date=1987&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D935868%23id-name%3DMR&rft.aulast=Dassios&rft.aufirst=George&rft.au=Kiriaki%2C+Kiriakie&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADouble+factorial" class="Z3988"></li> <li id="cite_note-22"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-22">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidt2010" class="citation journal cs1">Schmidt, Maxie D. 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Integers&rft.atitle=New+congruences+and+finite+difference+equations+for+generalized+factorial+functions&rft.volume=18&rft.pages=A78%3A1-A78%3A34&rft.date=2018&rft_id=info%3Aarxiv%2F1701.04741&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3862591%23id-name%3DMR&rft.aulast=Schmidt&rft.aufirst=Maxie+D.&rft_id=https%3A%2F%2Fmath.colgate.edu%2F~integers%2Fs78%2Fs78.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADouble+factorial" class="Z3988"></li> </ol> </div> </div>   </section> </div> <noscript> <img src="https://login.m.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&mobile=1" alt="" width="1" height="1" style="border: none; position: absolute;"> </noscript> <div class="printfooter" data-nosnippet=""> Retrieved from "<a dir="ltr" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/w/index.php?title%3DDouble_factorial%26oldid%3D1258451328">https://en.wikipedia.org/w/index.php?title=Double_factorial&oldid=1258451328</a>" </div> </div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"><a class="last-modified-bar" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Double_factorial&action=history&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB"> <div class="post-content last-modified-bar__content"> Last edited on 19 November 2024, at 20:26 </div></a> <div class="post-content footer-content"> <div id="mw-data-after-content"> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ar.wikipedia.org/wiki/%25D8%25B9%25D8%25A7%25D9%2585%25D9%2584%25D9%258A_%25D8%25AB%25D9%2586%25D8%25A7%25D8%25A6%25D9%258A" title="عاملي ثنائي – Arabic" lang="ar" hreflang="ar" data-title="عاملي ثنائي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/Doble_factorial" title="Doble factorial – Catalan" lang="ca" hreflang="ca" data-title="Doble factorial" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li> <li class="interlanguage-link interwiki-et mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.wikipedia.org/wiki/Kahekordne_faktoriaal" title="Kahekordne faktoriaal – Estonian" lang="et" hreflang="et" data-title="Kahekordne faktoriaal" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/Doble_factorial" title="Doble factorial – Spanish" lang="es" hreflang="es" data-title="Doble factorial" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li> <li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fa.wikipedia.org/wiki/%25D9%2581%25D8%25A7%25DA%25A9%25D8%25AA%25D9%2588%25D8%25B1%25DB%258C%25D9%2584_%25D8%25AF%25D9%2588%25D8%25A8%25D9%2584" title="فاکتوریل دوبل – Persian" lang="fa" hreflang="fa" data-title="فاکتوریل دوبل" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li> <li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/wiki/Analogues_de_la_factorielle%23Multifactorielles" title="Analogues de la factorielle – French" lang="fr" hreflang="fr" data-title="Analogues de la factorielle" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li> <li class="interlanguage-link interwiki-he mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://he.wikipedia.org/wiki/%25D7%25A2%25D7%25A6%25D7%25A8%25D7%25AA_%25D7%259B%25D7%25A4%25D7%2595%25D7%259C%25D7%2594" title="עצרת כפולה – Hebrew" lang="he" hreflang="he" data-title="עצרת כפולה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Dubbelfaculteit" title="Dubbelfaculteit – Dutch" lang="nl" hreflang="nl" data-title="Dubbelfaculteit" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li> <li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikipedia.org/wiki/%25E4%25BA%258C%25E9%2587%258D%25E9%259A%258E%25E4%25B9%2597" title="二重階乗 – Japanese" lang="ja" hreflang="ja" data-title="二重階乗" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li> <li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pt.wikipedia.org/wiki/Duplo_fatorial" title="Duplo fatorial – Portuguese" lang="pt" hreflang="pt" data-title="Duplo fatorial" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li> <li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%2594%25D0%25B2%25D0%25BE%25D0%25B9%25D0%25BD%25D0%25BE%25D0%25B9_%25D1%2584%25D0%25B0%25D0%25BA%25D1%2582%25D0%25BE%25D1%2580%25D0%25B8%25D0%25B0%25D0%25BB" title="Двойной факториал – Russian" lang="ru" hreflang="ru" data-title="Двойной факториал" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li> <li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://simple.wikipedia.org/wiki/Double_factorial" title="Double factorial – Simple English" lang="en-simple" hreflang="en-simple" data-title="Double factorial" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li> <li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sk.wikipedia.org/wiki/Dvojit%25C3%25BD_faktori%25C3%25A1l" title="Dvojitý faktoriál – Slovak" lang="sk" hreflang="sk" data-title="Dvojitý faktoriál" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Semifakultet" title="Semifakultet – Swedish" lang="sv" hreflang="sv" data-title="Semifakultet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li> <li class="interlanguage-link interwiki-uk badge-Q70893996 mw-list-item" title=""><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B4%25D0%25B2%25D1%2596%25D0%25B9%25D0%25BD%25D0%25B8%25D0%25B9_%25D1%2584%25D0%25B0%25D0%25BA%25D1%2582%25D0%25BE%25D1%2580%25D1%2596%25D0%25B0%25D0%25BB" title="Подвійний факторіал – Ukrainian" lang="uk" hreflang="uk" data-title="Подвійний факторіал" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li> <li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh-yue.wikipedia.org/wiki/%25E9%259B%2599%25E9%259A%258E%25E4%25B9%2598" title="雙階乘 – Cantonese" lang="yue" hreflang="yue" data-title="雙階乘" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li> </ul> </section> </div> <div class="minerva-footer-logo"> <img 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Double factorial - Wikipedia