Double factorial

In mathematics, the double factorial, or semifactorial, n‼ of a positive integer n is the product of all the positive integers up to n that have the same parity (odd or even) as n. That is, n ! ! = ∏ k = 0 ⌈ n 2 ⌉ − 1 ( n − 2 k ) = n ( n − 2 ) ( n − 4 ) ⋯ . {\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 is n ! ! = ∏ k = 1 n 2 ( 2 k ) = n ( n − 2 ) ( n − 4 ) ⋯ 4 ⋅ 2 , {\displaystyle n!!=\prod _{k=1}^{\frac {n}{2}}(2k)=n(n-2)(n-4)\cdots 4\cdot 2\,,} while for odd n it is n ! ! = ∏ k = 1 n + 1 2 ( 2 k − 1 ) = n ( n − 2 ) ( n − 4 ) ⋯ 3 ⋅ 1 . {\displaystyle n!!=\prod _{k=1}^{\frac {n+1}{2}}(2k-1)=n(n-2)(n-4)\cdots 3\cdot 1\,.} A few examples are: 1!! = 1 = 1, 2!! = 2 = 2, 3!! = 3 × 1 = 3, 4!! = 4 × 2 = 8, 5!! = 5 × 3 × 1 = 15, 6!! = 6 × 4 × 2 = 48, 7!! = 7 × 5 × 3 × 1 = 105. Often, both 0!! and (−1)!! are considered empty products that evaluate to 1. 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. There is another definition of double factorial, which can be evaluated...