Double factorial
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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,
Restated, this says that for even n, the double factorial is
while for odd n it is
For example, 9‼ = 9 × 7 × 5 × 3 × 1 = 945. The zero double factorial 0‼ = 1 as an empty product.[2][3]
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.[4][5]
The term semifactorial is also used by Knuth as a synonym of double factorial.[6]