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 .

*This was last updated in*September 2005

*Posted by:*Margaret Rouse

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