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,

$$n!!=n(n-2)(n-4)\mathrm{\cdots }{k}_n$$

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