## Double Factorial

The double factorial is a generalization of the usual Factorial defined by

 (1)

For , 1, 2, ..., the first few values are 1, 1, 2, 3, 8, 15, 48, 105, 384, ... (Sloane's A006882).

There are many identities relating double factorials to Factorials. Since
 (2)
it follows that . Since

 (3)

it follows that . Since
 (4)
it follows that
 (5)

Similarly, for , 1, ...,
 (6)

For Odd,
 (7)

For Even,
 (8)

Therefore, for any ,
 (9)

 (10)

The double factorial is a special case of the Multifactorial.

Sloane, N. J. A. Sequence A006882/M0876 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.