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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
n!!  :=
n
i  = 1
  
[in  (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!! xn  =  ?.
The exponential generating function for
n!!
is
E{n!!}(x)  :=
n  = 0
  
n!!
xn
n!
 =  1 + e   x 2 / 2 (1 +  
2  
 π  
 2 
erf  (
x
2  2 
)),

where

erf (z)  :=
 2
2  π 
z
0
e  − t  2dt  = 
 2
2  π 
  
n  = 0
  
(−1)n z 2 n  + 1
n!  (2 n + 1)

is the error function (erf  ).[1]

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)!!  =  2n  n!, n ≥ 0.
A000165 Double factorial of even numbers:
(2 n)!!  =  2n  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)!! xn  =  ?.
The exponential generating function for
(2 n)!!
is
E{(2 n)!!}(x)  :=
n  = 0
  
(2 n)!!
xn
n!
 =  E{2n  n!}(x)  = 
1
1 − 2 x
 = 
n  = 0
  
(2 x)n  = 
n  = 0
  
2n  n!
xn
n!
 = 
n  = 0
  
(2 n)!!
xn
n!
 .

Note the following Maclaurin series expansion

2  1 + sin x
 = 
n  = 0
  
(−1)
⌊  n / 2⌋
(2 n)!!
xn  =  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
2n  n!
 =  {
n  = 0
  
xn
n!
}x =  
1
 2
 =  {ex}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)!
2n  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)!! xn  =  ?.
The exponential generating function for
(2n + 1)!!
is
E{(2 n  + 1)!!}(x)  :=
n   = 0
  
(2 n + 1)!!
xn
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
  
2n  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
2n  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[2]

     
((
n
r
)):=
n!!
(nr)!! 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
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.
  2. 2.0 2.1 http://www.maa.org/pubs/mag_jun12_toc.html