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

n!!  :=    n ∏   i  = 1    [i ≡ n  (mod 2)] i, n ≥ 0,

where

[·]

is the Iverson bracket, and where for

n = 0

we get the empty product, i.e. 1.

Alternatively, we have

     

0!!  :=  1, 1!!  :=  1, n!!  :=    ⌊  n / 2⌋  −  1 ∏   i  = 0    (n − 2 i), n ≥ 2.

The double factorial of nonnegative integers is defined recursively as

     

0!!  :=  1, 1!!  :=  1, n!!  :=  n  ⋅  (n − 2)!!, n ≥ 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

G{n!!}(x)  :=    ∞ ∑   n  = 0    n!! x n  =  ?.

The exponential generating function for

n!!

is

E{n!!}(x)  :=    ∞ ∑   n  = 0    n!!  x n  n!  =  1 + e   x 2 / 2 (1 +   2√    π    2    erf  ( x 2√  2   )),

where

erf (z)  :=   2 2√  π  ∫  z 0 e  − t  2d t  =   2 2√  π       ∞ ∑   n  = 0    (−1) n z 2 n  + 1 n!  (2 n + 1)

is the error function (erf  ).

A [generalized] continued fraction generating function for

n!!

is
     

C{n!!}(x)  =  ? +  ?  ? +  ?  ? +  ?  ? +  ?  ? +  ?  ? +  ?  ? +  ?  ? +  ?  ? +  ?  ? +  ? ⋱ , n ≥ 0.

Sum of reciprocals of double factorial of nonnegative integers

∞ ∑   n  = 0    1 n!!  =    ∞ ∑   n  = 0    { 1 (2 n)!! + 1 (2 n + 1)!! }  =    ∞ ∑   n  = 0    1 (2 n)!!  +    ∞ ∑   n  = 0    1 (2 n + 1)!!  =

2√  e  +  2√  e   ∞ ∑   n  = 0    (−1) n (2 n)!! (2 n + 1)  =  2√  e {1 +   ∞ ∑   n  = 0    (−1) n (2 n)!! (2 n + 1) }.

Double factorial of even nonnegative integers

The double factorial of even nonnegative integers is given by

(2 n)!!  =  2 n  n!, n ≥ 0.

A000165 Double factorial of even numbers:

(2 n)!!  =  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 (2 n)!!

The generating function for

(2 n)!!

is

G{(2 n)!!}(x)  :=    ∞ ∑   n  = 0    (2 n)!! x n  =  ?.

The exponential generating function for

(2 n)!!

is

E{(2 n)!!}(x)  :=    ∞ ∑   n  = 0    (2 n)!!  x n  n!  =  E{2 n  n!}(x)  =  1 1 − 2 x  =

∞ ∑   n  = 0    (2 x) n  =    ∞ ∑   n  = 0    2 n  n!  x n  n!  =    ∞ ∑   n  = 0    (2 n)!!  x n  n!  .

Note the following Maclaurin series expansion

2√  1 + sin x  =    ∞ ∑   n  = 0    (−1)  ⌊  n / 2⌋ (2 n)!! x n  =  1 + 1 2 x − 1 8 x 2 − 1 48 x 3 + 1 384 x 4 + 1 3840 x 5 − 1 46080 x 6 − 1 645120 x 7 + ⋯.

A [generalized] continued fraction generating function for

(2 n)!!

is
     

C{(2 n)!!}(x)  =  1 1 −  2 x 1 −  2 x 1 −  4 x 1 −  4 x 1 −  6 x 1 −  6 x 1 −  8 x 1 −  8 x ⋱ , n ≥ 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

∞ ∑   n  = 0    1 (2 n)!!  =    ∞ ∑   n  = 0    1 2 n  n!  =  {   ∞ ∑   n  = 0     x n  n! }x =   1  2  =  {e x}x =   1  2  =  2√  e  .

Double factorial of odd nonnegative integers

The double factorial of odd nonnegative integers is given by

(2 n + 1)!!  =  (2 n + 1) (2 n − 1)!!  =  (2 n + 1) (2 n)! (2 n)!!  =  (2 n + 1)! (2 n)!!  =  (2 n + 1)! 2 n  n! , n ≥ 0.

A001147 Double factorial of odd numbers:

(2 n  −  1)!! = 1 ⋅  3 ⋅  5 ⋅  ... ⋅  (2 n  −  1), n   ≥   1

.

{1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425, 654729075, 13749310575, 316234143225, 7905853580625, 213458046676875, 6190283353629375, 191898783962510625, ...}

Generating functions for (2 n + 1)!!

The generating function for

(2n + 1)!!

is

G{(2 n  + 1)!!}(x)  :=    ∞ ∑   n   = 0    (2 n + 1)!! x n  =  ?.

The exponential generating function for

(2n + 1)!!

is

E{(2 n  + 1)!!}(x)  :=    ∞ ∑   n   = 0    (2 n + 1)!!  x n  n!  =  1 2√  1 − 2 x  .

A [generalized] continued fraction generating function for

(2n + 1)!!

is
     

C{(2 n  + 1)!!}(x)  =  1 1 −  3 x 1 −  2 x 1 −  5 x 1 −  4 x 1 −  7 x 1 −  6 x 1 −  9 x 1 −  8 x ⋱ , n ≥ 0.

Sum of reciprocals of double factorial of odd nonnegative integers

∞ ∑   n   = 0    1 (2 n + 1)!!  =    ∞ ∑   n   = 0    2 n  n! (2 n + 1)!  =    ∞ ∑   n   = 0    (2 n)!! (2 n)! (2 n + 1)  =

2√   π  e 2   erf  ( 1 2√  2  )  =  2√  e    ∞ ∑   n   = 0    (−1) n 2 n  n! (2 n + 1)  =  2√  e    ∞ ∑   n   = 0    (−1) n (2 n)!! (2 n + 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

∞ ∑   n   = 0    1 (2 n + 1)!!  =  1.410686134642447997690824711419115041323478....

A060196 Decimal expansion of

∑ ∞ k   = 0 1 (2 k + 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

     

(( n r ))  :=  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

${\begin{array}{l}\displaystyle {n\underbrace {!\cdots \,!} _{k~{\text{times}}}:=\prod _{i=1}^{n}\,[i\equiv n{\pmod {k}}]\,i,\quad n\geq 0,}\end{array}}$

where

[·]

is the Iverson bracket, and where for

n = 0

we get the empty product, i.e. 1.

Alternatively, we have

${\begin{array}{l}\displaystyle {\begin{aligned}0\underbrace {!\cdots \,!} _{k~{\text{times}}}&:=1,\\n\underbrace {!\cdots \,!} _{k~{\text{times}}}&:=n,\quad 1\leq n\leq k-1,\\n\underbrace {!\cdots \,!} _{k~{\text{times}}}&:=\prod _{i=0}^{\left\lfloor n/k\right\rfloor -1}(n-k\,i),\quad n\geq k.\end{aligned}}\end{array}}$

The multifactorial of nonnegative integers is defined recursively as

${\begin{array}{l}\displaystyle {\begin{aligned}0\underbrace {!\cdots \,!} _{k~{\text{times}}}&:=1,\\n\underbrace {!\cdots \,!} _{k~{\text{times}}}&:=n,\quad 1\leq n\leq k-1,\\n\underbrace {!\cdots \,!} _{k~{\text{times}}}&:=n\cdot (n-k)\underbrace {!\cdots \,!} _{k~{\text{times}}},\quad n\geq k.\end{aligned}}\end{array}}$

Multifactorials

k	$\textstyle {n\underbrace {!\cdots \,!} _{k~{\text{times}}},\,n\geq 0}$	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

${\begin{array}{l}\displaystyle {\underbrace {\left(\cdots \left({n \atop r}\right)\cdots \right)} _{k~{\text{times}}}:={\frac {n\underbrace {!\cdots \,!} _{k~{\text{times}}}}{(n-r)\underbrace {!\cdots \,!} _{k~{\text{times}}}\;r\underbrace {!\cdots \,!} _{k~{\text{times}}}}}.}\end{array}}$

See also

• Factorial