Factors of binomial sums from the Catalan triangle

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 n1,…,nmn1,…,nm, nm+1=n1nm+1=n1, and any nonnegative integer r, the expressionn1−1(n1+nmn1)−1∑k=1n1k2r+1∏i=1m(ni+ni+1ni+k) is either an integer or a half-integer. Moreover, several related conjectures are proposed.