The double factorial, symbolized by two exclamation marks (!!), is a quantity defined for all integer s greater than or equal to -1. For an even integer n , the double factorial is the product of all even integers less than or equal to n but greater than or equal to 2. For an odd integer p , the double factorial is the product of all odd integers less than or equal to p and greater than or equal to 1. The double factorial values of 0 and -1 are defined as equal to 1. Double factorial values for integers less than -1 are not defined.
Mathematically, the formulas for double factorial are as follows. If n is even, then
n !! = n ( n - 2)( n - 4)( n - 6) ... (4)(2)
If p is odd, then
p !! = p ( p - 2)( p - 4)( p - 6) ... (3)(1)
If q = 0 or q = -1, then q !! = 1 by convention.
The double factorial is mainly of interest to number theorists. It occasionally arises in statistics, combinatorics, calculus, and physics. Compare factorial .