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| {{Noref|date=August 2011}}
| | Nice to satisfy you, I am Marvella Shryock. Puerto Rico is exactly where he's usually been residing but she requirements to move simply because of her family. Doing ceramics is what love doing. Managing people has been his working day occupation for a while.<br><br>My blog :: [http://www.pponline.co.uk/user/miriamlinswkucrd over the counter std test] |
| '''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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Nice to satisfy you, I am Marvella Shryock. Puerto Rico is exactly where he's usually been residing but she requirements to move simply because of her family. Doing ceramics is what love doing. Managing people has been his working day occupation for a while.
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