Factors of binomial sums from the Catalan triangle

Details Factors of binomial sums from the Catalan triangle Factors of binomial sums from the Catalan triangle

2009-09-02

arxiv.org/abs/0909.0307v2

J. Number Theory 130 (2010), 172--186

Mathematics

Number Theory

15 pages, final version

Scientific paper

10.1016/j.jnt.2009.07.005

By using the Newton interpolation formula, we generalize the recent identities on the Catalan triangle obtained by Miana and Romero as well as those of Chen and Chu. We further study divisibility properties of sums of products of binomial coefficients and an odd power of a natural number. For example, we prove that for all positive integers $n_1, ..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, the expression $$n_1^{-1}{n_1+n_{m}\choose n_1}^{-1} \sum_{k=1}^{n_1}k^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}\choose n_i+k}$$ is either an integer or a half-integer. Moreover, several related conjectures are proposed.

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