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| {{Noref|date=August 2011}}
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| '''Jacques Touchard''' (1885 – 1968) was a [[French people|French]] [[mathematician]]. In 1953, he proved that an odd [[perfect number]] must be of the form 12''k'' + 1 or 36''k'' + 9. In [[combinatorics]] and [[probability theory]], he introduced the [[Touchard polynomials]]. He is also known for his solution to the [[ménage problem]] of counting seating arrangements in which men and women alternate and are not seated next to their spouses.
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| ==Touchard's Catalan identity==
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| The following algebraic identity involving the [[Catalan number]]s
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| :<math> C_k ={ 1\over{k+1}}{{2k}\choose {k}},\quad k \ge 0 </math>
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| is apparently due to Touchard (according to [[Richard P. Stanley]], who mentions it in his panorama article "[http://www-math.mit.edu/~rstan/ec/catalan.pdf Exercises on Catalan and Related Numbers]" giving an overwhelming plenitude of different definitions for the Catalan numbers).
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| For ''n'' ≥ 0 one has
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| :<math> C_{n+1} =\sum_{k \,\le\, n/2} 2^{n-2k} {n \choose 2k} C_k. \,</math>
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| Using the [[generating function]]
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| :<math> C(t)=\sum_{n \ge 0} C_n t^n ={{1-\sqrt{1-4t}}\over {2t}} </math>
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| it can be proved by algebraic manipulations of generating [[series (mathematics)|series]] that Touchard's identity is
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| equivalent to the [[functional equation]]
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| :<math> {t \over {1-2t}} C\left({t^2\over (1-2t)^2}\right) = C(t)-1 </math>
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| satisfied by the Catalan generating series ''C''(''t'').
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| {{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
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| | NAME = Touchard, Jacques
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| | ALTERNATIVE NAMES =
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| | SHORT DESCRIPTION = French mathematician
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| | DATE OF BIRTH = 1885
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| | PLACE OF BIRTH =
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| | DATE OF DEATH = 1968
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| | PLACE OF DEATH =
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| }}
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| {{DEFAULTSORT:Touchard, Jacques}}
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| [[Category:French mathematicians]]
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| [[Category:1885 births]]
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| [[Category:1968 deaths]]
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| [[Category:Place of birth missing]]
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| {{france-mathematician-stub}}
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