Double Factorial

Date: 02/22/2002 at 03:06:42
From: Linda Mitchell
Subject: Factorial !!

Hi Dr Math.

Please can you tell me what TWO ! marks mean in factorial questions?

For example (1/.2)!! is an expression I have seen, but while I 
understand that the first ! would mean 5*4*3*2*1, I have no idea what 
the second ! does to it.

Thank you,
Linda M.

Date: 02/22/2002 at 04:03:35
From: Doctor Pete
Subject: Re: Factorial !!

Hi Linda,

The notation of two factorials, e.g.,

     x!!

is a single symbol, called the double factorial.  It is the product of 
every other positive integer less than or equal to x, so for instance,

     9!! = 9*7*5*3*1,
and
     12!! = 12*10*8*6*4*2.

This is not to be confused with a repeated factorial, which would be 
written

     (x!)!,

where we would have for the case x = 5, (5!)! = 120!, a much larger 
number than 5!! = 15.

It is not too difficult to show that if n is even, then

     n!! = (2^(n/2))(n/2)!,

and if n is odd, then

     n!! = n!/(n-1)!! = n!/(2^((n-1)/2)((n-1)/2)!);

thus revealing a relation between the double and single factorial 
functions.

- Doctor Pete, The Math Forum
  http://mathforum.org/dr.math/