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# Double factorial

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The double factorial (sometimes called the semifactorial) of a nonnegative integer
 n
is defined as the product of positive integers having the same parity as
 n
${\displaystyle n!!:=\prod _{i=1}^{n}[i\equiv n{\pmod {2}}]\,i,\quad n\geq 0,}$
where
 [·]
is the Iverson bracket, and where for
 n = 0
we get the empty product, i.e.
 1
.

Alternatively, we have

${\displaystyle 0!!:=1,}$
${\displaystyle 1!!:=1,}$
${\displaystyle n!!:=\prod _{i=0}^{\left\lfloor {\tfrac {n}{2}}\right\rfloor -1}(n-2i),\quad n\geq 2.}$

The double factorial of nonnegative integers is defined recursively as

${\displaystyle 0!!:=1,}$
${\displaystyle 1!!:=1,}$
${\displaystyle n!!:=n\cdot (n-2)!!,\quad n\geq 2.}$
A006882 Double factorials
 n!!
:
 a (0) = a (1) = 1; a (n) = n  ⋅   a (n  −  2), n   ≥   2
.
 {1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, 10395, 46080, 135135, 645120, 2027025, 10321920, 34459425, 185794560, 654729075, 3715891200, 13749310575, 81749606400, 316234143225, ...}

## Generating functions for n!!

The generating function for
 n!!
is
${\displaystyle G_{\{n!!\}}(x)\equiv \sum _{n=0}^{\infty }n!!\,x^{n}=\ ?.}$
The exponential generating function for
 n!!
is
${\displaystyle E_{\{n!!\}}(x)\equiv \sum _{n=0}^{\infty }n!!\,{\frac {x^{n}}{n!}}=1+e^{\frac {x^{2}}{2}}~{\Big (}1+{\sqrt {\tfrac {\pi }{2}}}~{\rm {erf}}\left({\tfrac {x}{\sqrt {2}}}\right){\Big )},}$

where

${\displaystyle {\rm {erf}}(z):={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}dt={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}\,z^{2n+1}}{n!\,(2n+1)}}}$

is the error function (erf).[1]

A [generalized] continued fraction generating function for
 n!!
is
${\displaystyle C_{\{n!!\}}(x)={\cfrac {?}{?-{\cfrac {?}{?-{\cfrac {?}{?-{\cfrac {?}{?-{\cfrac {?}{?-{\cfrac {?}{?-{\cfrac {?}{?-{\cfrac {?}{?-{\cfrac {?}{\ddots }}}}}}}}}}}}}}}}}},\quad n\geq 0.}$

## Sum of reciprocals of double factorial of nonnegative integers

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n!!}}=\sum _{n=0}^{\infty }{\Bigg \{}{\frac {1}{(2n)!!}}+{\frac {1}{(2n+1)!!}}{\Bigg \}}=\sum _{n=0}^{\infty }{\frac {1}{(2n)!!}}~+~\sum _{n=0}^{\infty }{\frac {1}{(2n+1)!!}}={\sqrt {e}}~+~{\sqrt {e}}\,\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!!\,(2n+1)}}={\sqrt {e}}~{\Bigg \{}1+\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!!\,(2n+1)}}{\Bigg \}}}$

## Double factorial of even nonnegative integers

The double factorial of even nonnegative integers is given by

${\displaystyle (2n)!!=2^{n}n!,\quad n\geq 0.}$
A000165 Double factorial of even numbers:
 (2n)!! = 2 n n!, n   ≥   0
.
 {1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560, 3715891200, 81749606400, 1961990553600, 51011754393600, 1428329123020800, 42849873690624000, 1371195958099968000, ...}

### Generating functions for (2n)!!

The generating function for
 (2n)!!
is
${\displaystyle G_{\{(2n)!!\}}(x)\equiv \sum _{n=0}^{\infty }(2n)!!\,x^{n}=\ ?.}$
The exponential generating function for
 (2n)!!
is
${\displaystyle E_{\{(2n)!!\}}(x)\equiv \sum _{n=0}^{\infty }(2n)!!\,{\frac {x^{n}}{n!}}=E_{\{2^{n}n!\}}(x)={\frac {1}{1-2x}}=\sum _{n=0}^{\infty }(2x)^{n}=\sum _{n=0}^{\infty }2^{n}n!{\frac {x^{n}}{n!}}=\sum _{n=0}^{\infty }(2n)!!{\frac {x^{n}}{n!}}.}$

Note the following Maclaurin series expansion

${\displaystyle {\sqrt {1+\sin(x)}}=\sum _{n=0}^{\infty }{\frac {(-1)^{\lfloor {\frac {n}{2}}\rfloor }}{(2n)!!}}x^{n}=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}-{\frac {1}{48}}x^{3}+{\frac {1}{384}}x^{4}+{\frac {1}{3840}}x^{5}-{\frac {1}{46080}}x^{6}-{\frac {1}{645120}}x^{7}+\ldots }$
A [generalized] continued fraction generating function for
 (2n)!!
is
${\displaystyle C_{\{(2n)!!\}}(x)={\cfrac {1}{1-{\cfrac {2x}{1-{\cfrac {2x}{1-{\cfrac {4x}{1-{\cfrac {4x}{1-{\cfrac {6x}{1-{\cfrac {6x}{1-{\cfrac {8x}{1-{\cfrac {8x}{\ddots }}}}}}}}}}}}}}}}}},\quad n\geq 0.}$

### Sum of reciprocals of double factorial of even nonnegative integers

The sum of reciprocals of double factorial of even nonnegative integers equals
 2√  e
, since
${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{(2n)!!}}=\sum _{n=0}^{\infty }{\frac {1}{2^{n}~n!}}={\Bigg \{}\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}{\Bigg \}}_{x={\frac {1}{2}}}={\Big \{}e^{x}{\Big \}}_{x={\frac {1}{2}}}={\sqrt {e}}.}$

## Double factorial of odd nonnegative integers

The double factorial of odd nonnegative integers is given by

${\displaystyle (2n+1)!!=(2n+1)(2n-1)!!=(2n+1){\frac {(2n)!}{(2n)!!}}={\frac {(2n+1)!}{(2n)!!}}={\frac {(2n+1)!}{2^{n}~n!}},\quad n\geq 0.}$
A001147 Double factorial of odd numbers:
 (2n  −  1)!! = 1  ⋅   3  ⋅   5  ⋅   ...  ⋅   (2n  −  1), n   ≥   1
.
 {1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425, 654729075, 13749310575, 316234143225, 7905853580625, 213458046676875, 6190283353629375, 191898783962510625, ...}

### Generating functions for (2n + 1)!!

The generating function for
 (2n + 1)!!
is
${\displaystyle G_{\{(2n+1)!!\}}(x)\equiv \sum _{n=0}^{\infty }(2n+1)!!\,x^{n}=\ ?.}$
The exponential generating function for
 (2n + 1)!!
is
${\displaystyle E_{\{(2n+1)!!\}}(x)\equiv \sum _{n=0}^{\infty }(2n+1)!!\,{\frac {x^{n}}{n!}}={\frac {1}{\sqrt {1-2x}}}.}$
A [generalized] continued fraction generating function for
 (2n + 1)!!
is
${\displaystyle C_{\{(2n+1)!!\}}(x)={\cfrac {1}{1-{\cfrac {3x}{1-{\cfrac {2x}{1-{\cfrac {5x}{1-{\cfrac {4x}{1-{\cfrac {7x}{1-{\cfrac {6x}{1-{\cfrac {9x}{1-{\cfrac {8x}{\ddots }}}}}}}}}}}}}}}}}},\quad n\geq 0.}$

### Sum of reciprocals of double factorial of odd nonnegative integers

${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{(2n+1)!!}}=\sum _{n=0}^{\infty }{\frac {2^{n}~n!}{(2n+1)!}}=\sum _{n=0}^{\infty }{\frac {(2n)!!}{(2n)!~(2n+1)}}={\sqrt {\frac {\pi e}{2}}}\,{\rm {~erf}}({\tfrac {1}{\sqrt {2}}})={\sqrt {e}}\,\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2^{n}\,n!\,(2n+1)}}={\sqrt {e}}\,\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!!\,(2n+1)}}}$
This is the power series part of
2
 π e 2
obtained from the remarkable formula of Ramanujan evaluated at
 x = 1
. The decimal expansion (which is pretty close to
 2√  2 = 1.414213562373095...
, see A002193) is
${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{(2n+1)!!}}=1.410686134642447997690824711419115041323478\ldots .}$
A060196 Decimal expansion of
 ∞

 k   = 0
 1 (2k + 1)!!
= 1 +
 1 1 ⋅  3
+
 1 1 ⋅  3 ⋅  5
+
 1 1 ⋅  3 ⋅  5 ⋅  7
+ ...
 {1, 4, 1, 0, 6, 8, 6, 1, 3, 4, 6, 4, 2, 4, 4, 7, 9, 9, 7, 6, 9, 0, 8, 2, 4, 7, 1, 1, 4, 1, 9, 1, 1, 5, 0, 4, 1, 3, 2, 3, 4, 7, 8, 6, 2, 5, 6, 2, 5, 1, 9, 2, 1, 9, 7, 7, 2, 4, 6, 3, 9, 4, 6, 8, 1, 6, 4, 7, 8, 1, 7, 9, 8, 4, 9, 0, 3, 9, ...}

## Double factorial binomial coefficients

The double factorial binomial coefficients are[2]

${\displaystyle \left(\!\!\left({{n} \atop {r}}\right)\!\!\right):={\frac {n!!}{(n-r)!!\,r!!}}.}$

## Multifactorial

The
 k
-multifactorial of a nonnegative integer
 n
is defined as the product of positive integers having the same congruence
 (mod k)
as
 n
${\displaystyle n\!\!\underbrace {!\cdots !} _{k{\rm {~times}}}:=\prod _{i=1}^{n}[i\equiv n{\pmod {k}}]\,i,\quad n\geq 0,}$
where
 [·]
is the Iverson bracket, and where for
 n = 0
we get the empty product, i.e.
 1
.

Alternatively, we have

${\displaystyle 0\!\!\underbrace {!\cdots !} _{k{\rm {~times}}}:=1,}$
${\displaystyle n\!\!\underbrace {!\cdots !} _{k{\rm {~times}}}:=n,\quad 1\leq n\leq k-1,}$
${\displaystyle n\!\!\underbrace {!\cdots !} _{k{\rm {~times}}}:=\prod _{i=0}^{\left\lfloor {\tfrac {n}{k}}\right\rfloor -1}(n-ki),\quad n\geq k.}$

The multifactorial of nonnegative integers is defined recursively as

${\displaystyle 0\!\!\underbrace {!\cdots !} _{k{\rm {~times}}}:=1,}$
${\displaystyle n\!\!\underbrace {!\cdots !} _{k{\rm {~times}}}:=n,\quad 1\leq n\leq k-1,}$
${\displaystyle n\!\!\underbrace {!\cdots !} _{k{\rm {~times}}}:=n\cdot (n-k)\!\!\underbrace {!\cdots !} _{k{\rm {~times}}},\quad n\geq k.}$
Multifactorials
 k
${\displaystyle \textstyle {n\!\!\underbrace {!\cdots !} _{k{\rm {~times}}},\ n\geq 0}}$ A-number
0 {1, 1, ?, ...} (Is it possible to generalize for k = 0?) A??????
1 {1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, ...} A000142
2 {1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, 10395, 46080, 135135, 645120, 2027025, 10321920, 34459425, 185794560, 654729075, 3715891200, 13749310575, 81749606400, ...} A006882
3 {1, 1, 2, 3, 4, 10, 18, 28, 80, 162, 280, 880, 1944, 3640, 12320, 29160, 58240, 209440, 524880, 1106560, 4188800, 11022480, 24344320, 96342400, 264539520, 608608000, ...} A007661
4 {1, 1, 2, 3, 4, 5, 12, 21, 32, 45, 120, 231, 384, 585, 1680, 3465, 6144, 9945, 30240, 65835, 122880, 208845, 665280, 1514205, 2949120, 5221125, 17297280, 40883535, 82575360, ...} A007662
5 {1, 1, 2, 3, 4, 5, 6, 14, 24, 36, 50, 66, 168, 312, 504, 750, 1056, 2856, 5616, 9576, 15000, 22176, 62832, 129168, 229824, 375000, 576576, 1696464, 3616704, 6664896, ...} A085157
6 {1, 1, 2, 3, 4, 5, 6, 7, 16, 27, 40, 55, 72, 91, 224, 405, 640, 935, 1296, 1729, 4480, 8505, 14080, 21505, 31104, 43225, 116480, 229635, 394240, 623645, 933120, 1339975, ...} A085158
7 {1, 1, 2, 3, 4, 5, 6, 7, 8, 18, 30, 44, 60, 78, 98, 120, 288, 510, 792, 1140, 1560, 2058, 2640, 6624, 12240, 19800, 29640, 42120, 57624, 76560, 198720, 379440, 633600, 978120, ...} A114799
8 {1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 33, 48, 65, 84, 105, 128, 153, 360, 627, 960, 1365, 1848, 2415, 3072, 3825, 9360, 16929, 26880, 39585, 55440, 74865, 98304, 126225, 318240, ...} A114800
9 {1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 36, 52, 70, 90, 112, 136, 162, 190, 440, 756, 1144, 1610, 2160, 2800, 3536, 4374, 5320, 12760, 22680, 35464, 51520, 71280, 95200, ...} A114806
10
11
12

### Multifactorial binomial coefficients

The multifactorial binomial coefficients are[2]

${\displaystyle \underbrace {\left(\!\cdots \!\left({{n} \atop {r}}\right)\!\cdots \!\right)} _{k{\rm {~times}}}:={\frac {n\!\!\underbrace {!\cdots !} _{k{\rm {~times}}}}{(n-r)\!\!\underbrace {!\cdots !} _{k{\rm {~times}}}\,r\!\!\underbrace {!\cdots !} _{k{\rm {~times}}}}}.}$

## Notes

1. Weisstein, Eric W., Erf, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Erf.html]
2. http://www.maa.org/pubs/mag_jun12_toc.html