Factors of binomial sums from the Catalan triangle

TL;DR

This paper generalizes identities related to the Catalan triangle using Newton interpolation, explores divisibility properties of binomial coefficient sums, and proposes several conjectures on these topics.

Contribution

It introduces a generalized framework for Catalan triangle identities and investigates divisibility properties of binomial sums involving odd powers.

Findings

Proves integrality or half-integrality of specific binomial sums.

Extends known identities on the Catalan triangle via Newton interpolation.

Presents several new conjectures related to divisibility and identities.

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 $$n_1^{-1}{n_1+n_{m}\choose n_1}^{-1} \sum_{k=1}^{n_1}k^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}\choose n_i+k}$$ is either an integer or a half-integer. Moreover, several related conjectures are proposed.