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05A10-DoubleFactorial.tex
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05A10-DoubleFactorial.tex
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\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{DoubleFactorial}
\pmcreated{2013-03-22 12:24:54}
\pmmodified{2013-03-22 12:24:54}
\pmowner{drini}{3}
\pmmodifier{drini}{3}
\pmtitle{double factorial}
\pmrecord{9}{32318}
\pmprivacy{1}
\pmauthor{drini}{3}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{05A10}
\endmetadata
% 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}
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%%%\usepackage{xypic}
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\begin{document}
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 $(2n+1)!! = \frac{(2n+1)!}{2^n n!}$.
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\end{document}