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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, 10395, 46080, 135135, 645120, ... (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.

### Sources Edit

1. Double Factorial from Wolfram MathWorld

Main article: Factorial
Multifactorials: Double factorial · Multifactorial
Falling and rising: Falling factorial · Rising factorial
Other mathematical variants: Alternating factorial · Hyperfactorial · q-factorial · Roman factorial · Subfactorial · Weak factorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial