info prev up next book cdrom email home

Double Factorial

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

\begin{displaymath}
n!!\equiv\cases{
n\cdot(n-2)\ldots 5\cdot 3\cdot 1 & $n$\ o...
...(n-2)\ldots 6\cdot 4\cdot 2 & $n$\ even\cr
1 & $n=-1, 0$.\cr}
\end{displaymath} (1)

For $n=0$, 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
$(2n+1)!!2^nn!$
$= [(2n+1)(2n-1)\cdots 1][2n][2(n-1)][2(n-2)]\cdots 2(1)$
$= [(2n+1)(2n-1)\cdots 1][2n(2n-2)(2n-4)\cdots 2]$
$= (2n+1)(2n)(2n-1)(2n-2)(2n-3)(2n-4)\cdots 2(1)$
$= (2n+1)!,$ (2)
it follows that $(2n+1)!! = {(2n+1)!\over 2^nn!}$. Since

$\displaystyle (2n)!!$ $\textstyle =$ $\displaystyle (2n)(2n-2)(2n-4)\cdots 2$  
  $\textstyle =$ $\displaystyle [2(n)][2(n-1)][2(n-2)]\cdots 2 = 2^nn!,$ (3)

it follows that $(2n)!! = 2^nn!$. Since
$(2n-1)!!2^nn! $
$= [(2n-1)(2n-3)\cdots 1][2n][2(n-1)][2(n-2)]\cdots 2(1)$
$= (2n-1)(2n-3)\cdots 1][2n(2n-2)(2n-4)\cdots 2]$
$= 2n(2n-1)(2n-2)(2n-3)(2n-4)\cdots 2(1)$
$= (2n)!,$ (4)
it follows that
\begin{displaymath}
(2n-1)!! = {(2n)!\over 2^nn!}.
\end{displaymath} (5)

Similarly, for $n=0$, 1, ...,
\begin{displaymath}
(-2n-1)!! = {(-1)^n\over (2n-1)!!} = {(-1)^n2^nn!\over (2n)!}.
\end{displaymath} (6)

For $n$ Odd,
$\displaystyle {n!\over n!!}$ $\textstyle =$ $\displaystyle {n(n-1)(n-2)\cdots (1)\over n(n-2)(n-4)\cdots (1)}$  
  $\textstyle =$ $\displaystyle (n-1)(n-3)\cdots (1) = (n-1)!!.$ (7)

For $n$ Even,
$\displaystyle {n!\over n!!}$ $\textstyle =$ $\displaystyle {n(n-1)(n-2)\cdots (2)\over n(n-2)(n-4)\cdots (2)}$  
  $\textstyle =$ $\displaystyle (n-1)(n-3)\cdots (2)= (n-1)!!.$ (8)

Therefore, for any $n$,
\begin{displaymath}
{n!\over n!!}=(n-1)!!
\end{displaymath} (9)


\begin{displaymath}
n!=n!!(n-1)!!.
\end{displaymath} (10)


The double factorial is a special case of the Multifactorial.

See also Factorial, Multifactorial


References

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.



info prev up next book cdrom email home

© 1996-9 Eric W. Weisstein
1999-05-24