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The double factorial is a version on the factorial, defined as $$n!! = n \cdot (n - 2) \cdot (n - 4) \cdot \ldots$$.[1] In other words, it is made by multiplying all the positive odd numbers up to n if n is odd, or multiplying all the positive even numbers up to n if n is even.

The first few values of $$n!!$$ for $$n$$ = 0, 1, 2, 3, ... are 1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, 10,395, 46,080, 135,135, 645,120, ... (OEIS A006882) The sum of the reciprocals of these numbers is 2.059407405342577...

When $$n$$ is even,  $$n!! = (\frac{n}{2})! 2^{(\frac{n}{2})}$$.

It should be noted that $$n!!$$ is not equivalent to $$(n!)!$$ (nested factorial). Double factorial actually grows slower than factorial.

The multifactorial of nonnegative integers is defined recursively as:

Multifactorials
k width="25" style="text-align:Template:Math A-number
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

Sources Edit

1. Double Factorial from Wolfram MathWorld
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