DoubleFactorial 2002-02-20 20:07:11 2005-07-26 23:01:52 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The \emph{double factorial} of a positive integer $n$ is the product $n!!$ of the positive integers less than or equal to $n$ that have the same parity as $n$, that is, \[n!! = n (n-2) (n-4)\cdots k_n\] where $k_n$ denotes $1$ if $n$ is an odd number and $2$ if $n$ is an even number. For example, \[ 7!! = 7 \cdot 5 \cdot 3 \cdot 1 = 105 \] \[ 10!! = 10\cdot 8\cdot 6\cdot 4\cdot 2 = 3840 \] Note that $n!!$ is not the same as $(n!)!$. Observe that $(2n)!! = 2^n n!$ and $(2n+1)!! = \frac{(2n+1)!}{2^n n!}$.