Double factorial

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 $n$ . That is,

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!!=\prod _{k=1}^{\frac {n}{2}}(2k)=n(n-2)(n-4)\cdots 4\cdot 2\,,

while for odd $n$ it is

n!!=\prod _{k=1}^{\frac {n+1}{2}}(2k-1)=n(n-2)(n-4)\cdots 3\cdot 1\,.

For example, $9‼ = 9 \times 7 \times 5 \times 3 \times 1 = 945$ . The zero double factorial $0‼ = 1$ as an empty product.

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

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

(sequence A000165 in the OEIS)

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

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

(sequence A001147 in the OEIS)

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

The term semifactorial is also used by Knuth as a synonym of double factorial.

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