Problemas Teoremas

Outubro 16, 2010

Problema do mês :: Problem of the month #7

Filed under: Combinatória,Combinatorics,Matemática,Matemática Discreta,Math,Problem,Problem Of The Month,Problema do mês,Problemas — Américo Tavares @ 6:11 pm
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pdmpom20101017


Mostre que :: Show that

\displaystyle\sum_{n=k+1}^{N}\dfrac{(-1)^{n+k}n}{\displaystyle\binom{n}{k}\displaystyle\binom{n+k}{k}}-\displaystyle\sum_{n=k+1}^{N}\dfrac{(-1)^{n+k}n}{\displaystyle\binom{n-1}{k}\displaystyle\binom{n-1+k}{k}}

=\dfrac{k}{\displaystyle\binom{2k}{k}}+\dfrac{(-1)^{N+k-1}k}{\displaystyle\binom{N}{k}\displaystyle\binom{N+k}{k}}

 

Soluções: até 8 Novembro 2010, via acltavares@sapo.pt ou caixa de comentários.

Solutions: until November 8, 2010, via acltavares@sapo.pt or comment box.

Dezembro 1, 2007

A note on a combinatorial identity related to the Apéry’s constant ζ(3)

Filed under: Caderno,Combinatória,Combinatorics,Demonstração,Matemática,Math,Proof — Américo Tavares @ 10:50 am
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pdf: included in Caderno (see “caderno” page)

Abstract. A purely combinatorial proof of an identity (that can be used to justify one step of the Apéry´s sequences formulas) is established. Equivalent formulas for the Apéry’s double sequences are also presented, as well as the numerical values of the first four lines of these sequences.

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