Double factorial

The double factorial ${\displaystyle n!!}$ is not the same as applying the factorial function twice ${\displaystyle (n!)!}$.

In mathematics, the double factorial of a number n, denoted by n‼, is the product of all the positive integers up to n that have the same parity (odd or even) as n.^[1] That is,

$${\displaystyle n!!=\prod _{k=0}^{\left\lceil {\frac {n}{2}}\right\rceil -1}(n-2k)=n(n-2)(n-4)\cdots .}$$

Restated, this says that for even n, the double factorial^[2] is $${\displaystyle n!!=\prod _{k=1}^{\frac {n}{2}}(2k)=n(n-2)(n-4)\cdots 4\cdot 2\,,}$$ while for odd n it is $${\displaystyle n!!=\prod _{k=1}^{\frac {n+1}{2}}(2k-1)=n(n-2)(n-4)\cdots 3\cdot 1\,.}$$ For example, 9‼ = 9 × 7 × 5 × 3 × 1 = 945. The zero double factorial 0‼ = 1 as an empty product.^[3][4]

The sequence of double factorials for even n = 0, 2, 4, 6, 8,... starts as

The sequence of double factorials for odd n = 1, 3, 5, 7, 9,... starts as

The term odd factorial is sometimes used for the double factorial of an odd number.^[5][6]

The term semifactorial is also used by Knuth as a synonym of double factorial.^[7]