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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
where
[·]
is the Iverson bracket, and where for
n = 0
we get the empty product, i.e.
1
.

Alternatively, we have

The double factorial of nonnegative integers is defined recursively as

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
The exponential generating function for
n!!
is

where

is the error function (erf).[1]

A [generalized] continued fraction generating function for
n!!
is

Sum of reciprocals of double factorial of nonnegative integers

Double factorial of even nonnegative integers

The double factorial of even nonnegative integers is given by

A000165 Double factorial of even numbers:
(2n)!! = 2nn!, 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
The exponential generating function for
(2n)!!
is

Note the following Maclaurin series expansion

A [generalized] continued fraction generating function for
(2n)!!
is

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

Double factorial of odd nonnegative integers

The double factorial of odd nonnegative integers is given by

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
The exponential generating function for
(2n + 1)!!
is
A [generalized] continued fraction generating function for
(2n + 1)!!
is

Sum of reciprocals of double factorial of odd nonnegative integers

This is the power series part of
2
πe
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
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]

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
where
[·]
is the Iverson bracket, and where for
n = 0
we get the empty product, i.e.
1
.

Alternatively, we have

The multifactorial of nonnegative integers is defined recursively as

Multifactorials
k
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]

See also

Notes

  1. Weisstein, Eric W., Erf, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Erf.html]
  2. 2.0 2.1 http://www.maa.org/pubs/mag_jun12_toc.html