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
See also
Main article: Factorial
Multifactorials: Double factorial · MultifactorialFalling and rising: Falling factorial · Rising factorial
Other mathematical variants: Alternating factorial · Hyperfactorial · q-factorial · Roman factorial · Subfactorial · Weak factorial · Primorial · Compositorial · Semiprimorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Nested Factorials: Tetorial · Petorial · Ectorial · Zettorial · Yottorial
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Tetrofactorial · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial