Summation formulae for binomial moments

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Abstract

By combining the telescoping method with an algebraic relation, four classes of binomial moments are examined:

$\sum_{k=1}^{n}( \pm 1)^{k-1}\binom{2 n}{n-k} k^{m} \quad \text { and } \quad \sum_{k=1}^{n}( \pm 1)^{k-1}\left[\begin{array}{c} 2 n \\ n-k \end{array}\right] k^{m} .$

Several explicit summation formulae are established with most of them having not appeared previously in the literature except for the first class corresponding to the “+” sign.

Keywords

binomial moment / central binomial coefficient / telescoping method

CHANGE HISTORY

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