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, $$ n!! = n (n-2) (n-4)\cdots k_ $$ 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 $(2n+1)!! = \frac{(2n+1)!}{2^n n!}$ .