Editing Double factorial (section)

{{Short description|Mathematical function}}
{{hatnote|The double factorial <math>n!!</math> is not the same as applying the [[factorial]] function twice <math>(n!)!</math>.}}[[File:Chord diagrams K6 matchings.svg|thumb|360px|The fifteen different [[Chord diagram (mathematics)|chord diagrams]] on six points, or equivalently the fifteen different [[perfect matching]]s on a six-vertex [[complete graph]].  These are counted by the double factorial {{math|15 {{=}} (6 − 1)‼}}.]]

In [[mathematics]], the '''double factorial''' of a number {{mvar|n}}, denoted by {{math|''n''‼}}, is the product of all the [[positive integer]]s up to {{mvar|n}} that have the same [[Parity (mathematics)|parity]] (odd or even) as {{mvar|n}}.<ref name="callan">{{cite arXiv|title=A combinatorial survey of identities for the double factorial|first=David|last=Callan|eprint=0906.1317|year=2009|class=math.CO}}</ref> That is,

<math display="block">n!! = \prod_{k=0}^{\left\lceil\frac{n}{2}\right\rceil - 1} (n-2k) = n (n-2) (n-4) \cdots.</math>

Restated, this says that for even {{mvar|n}}, the double factorial<ref>Some authors define the double factorial differently for even numbers; see {{slink|Double factorial|complex arguments}} below.</ref> is
<math display="block">n!! = \prod_{k=1}^\frac{n}{2} (2k) = n(n-2)(n-4)\cdots 4\cdot 2 \,,</math>
while for odd {{mvar|n}} it is
<math display="block">n!! = \prod_{k=1}^\frac{n+1}{2} (2k-1) = n(n-2)(n-4)\cdots 3\cdot 1 \,.</math>
For example, {{math|9‼ {{=}} 9 × 7 × 5 × 3 × 1 {{=}} 945}}. The zero double factorial {{math|0‼ {{=}} 1}} as an [[empty product]].<ref name=":0">{{Cite web|last=Weisstein|first=Eric W.|title=Double Factorial|url=https://mathworld.wolfram.com/DoubleFactorial.html|access-date=2020-09-10|website=mathworld.wolfram.com|language=en}}</ref><ref>{{Cite web|title=Double Factorials and Multifactorials {{!}} Brilliant Math & Science Wiki|url=https://brilliant.org/wiki/double-factorials-and-multifactorials/|access-date=2020-09-10|website=brilliant.org|language=en-us}}</ref>

The [[sequence]] of double factorials for even {{mvar|n}} = {{math|0, 2, 4, 6, 8,...}} starts as
{{block indent|{{math|1, 2, 8, 48, 384, 3840, 46080, 645120, ...}} {{OEIS|id=A000165}} }}
The sequence of double factorials for odd {{mvar|n}} = {{math|1, 3, 5, 7, 9,...}} starts as
{{block indent|{{math|1, 3, 15, 105, 945, 10395, 135135, ...}} {{OEIS|id=A001147}} }}

The term '''odd factorial''' is sometimes used for the double factorial of an odd number.<ref>{{cite journal
 | last1 = Henderson | first1 = Daniel J.
 | last2 = Parmeter | first2 = Christopher F.
 | doi = 10.1016/j.spl.2012.03.013
 | issue = 7
 | journal = Statistics & Probability Letters
 | mr = 2929790
 | pages = 1383–1387
 | title = Canonical higher-order kernels for density derivative estimation
 | volume = 82
 | year = 2012}}</ref><ref>{{cite journal
 | last = Nielsen | first = B.
 | doi = 10.1093/biomet/86.2.279
 | issue = 2
 | journal = Biometrika
 | mr = 1705359
 | pages = 279–288
 | title = The likelihood-ratio test for rank in bivariate canonical correlation analysis
 | volume = 86
 | year = 1999}}</ref>
<!--
The '''odd factorial''' of ''n'' is defined to be the product of all odd positive integers up to ''n'', that is, ''n''‼ if ''n'' is odd and (''n''−1)‼ if ''n'' is even.
-->

The term '''semifactorial''' is also used by [[Donald Knuth|Knuth]] as a synonym of double factorial.<ref>{{Cite book |last=Knuth |first=Donald Ervin |authorlink=Donald Knuth |title=The art of computer programming. volume 4B part 2: Combinatorial algorithms |date=2023 |publisher=Addison-Wesley |isbn=978-0-201-03806-4 |location=Boston Munich}}</ref>