Double factorial

The double factorial is a version on the factorial, defined as \(n!! = n \cdot (n - 2) \cdot (n - 4) \cdot \ldots\).Double Factorial from Wolfram MathWorld 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, 3,840, 10,395, 46,080, 135,135, 645,120, ... () 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, as there are always fewer than \(n\) arguments to multiply.
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zh:雙階乘
Category:Functions
Category:Factorials