  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.
