Double factorial

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

(A consequence of this definition is that 0!! = 1, as an empty product.)

Therefore, for even $n$ the double factorial is

and for odd $n$ it is

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

The double factorial should not be confused with the factorial function iterated twice, which is written as $(n!)!$ and not $n!!$

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

Merserve (1948) (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.

Because the double factorial only involves about half the factors of the ordinary factorial, its value is not substantially larger than the square root of the factorial $n!$ , and it is much smaller than the iterated factorial $(n!)!$ .

For an even positive integer $n = 2 k$ , $k \geq 0$ , the double factorial may be expressed as

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