double factorial

The double factorial of a positive integer $n$ is the product $n!!$ of the positive integers less than or equal to $n$ that have the same parity as $n$ , that is,

n!!=n(n-2)(n-4)\\cdots k_{n}

where $k_{n}$ denotes $1$ if $n$ is an odd number and $2$ if $n$ is an even number.

For example,

7!!=7\\cdot 5\\cdot 3\\cdot 1=105

10!!=10\\cdot 8\\cdot 6\\cdot 4\\cdot 2=3840

Note that $n!!$ is not the same as $(n!)!$ .

Observe that $(2n)!!=2^{n}n!$ and $(2n+1)!!=\\frac{(2n+1)!}{2^{n}n!}$ .