What is "double factorial"

n. (context mathematics combinatorics English) For an odd integer, the result of multiplying all the odd integers from 1 to the given number; or for an even integer, the result of multiplying all the even integers from 2 to the given number; symbolised by a double exclamation mark (!!). For example, 9!! = 1 * 3 * 5 * 7 * 9 = 945.

In mathematics, the product of all the integers from 1 up to some non-negative integer n that have the same parity as n is called the double factorial or semifactorial of n and is denoted by n!!. That is,

n!! = ∏(n − 2k) = n(n − 2)(n − 4)⋯
where m = ⌈n/2⌉ − 1. A consequence of this definition is that (as an empty product)

0!! = 1.
The double factorial should not be confused with the factorial function iterated twice, which is written as (x!)! and not x!!

For example, 9!! = 1 × 3 × 5 × 7 × 9 = 945.

For even n the double factorial is

n!! = ∏(2k) = n(n − 2)⋯2.
For odd n it is

n!! = ∏(2k − 1) = n(n − 2)⋯1.

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

1, 2, 8, 48, 384, 3840, 46080, 645120, ....

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

1, 3, 15, 105, 945, 10395, 135135, ....

(possibly the earliest publication to use double factorial notation) states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals arising in the derivation of the Wallis product. Double factorials also arise in expressing the volume of a hypersphere, and they have many applications in enumerative combinatorics.

The term odd factorial is sometimes used for the double factorial of an odd number.