Factors of binomial sums from the Catalan triangle

Abstract

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 $$n_1^{-1}{\left(\genfrac{}{}{0ex}{}{n_1+n_m}{n_1}\right)}^{-1}\sum _{k=1}^{n_1}{k}^{2r+1}\prod _{i=1}^m\left(\genfrac{}{}{0ex}{}{n_i+n_{i+1}}{n_i+k}\right)$$ is either an integer or a half-integer. Moreover, several related conjectures are proposed.

15 pages, final version

References in corpus (1)

• Binomial coefficients and the ring of p-adic integers

Cited by in corpus (10)

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• Factors of sums and alternating sums involving binomial coefficients and powers of integers
• Generalising Tuenter's binomial sums
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• Revitalized automatic proofs: demonstrations
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• Proof of a conjecture related to divisibility properties of binomial coefficients
• Factors of sums and alternating sums of products of $q$-binomial coefficients and powers of $q$-integers
• Combinatorial identities related to $2\times 2$ submatrices of recursive matrices
• On 2-adic orders of some binomial sums