  We study divisibility properties of certain sums and alternating sums
involving binomial coefficients and powers of integers. For example, we prove
that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any
nonnegative integer $r$, there holds {align*} \sum_{k=0}^{n_1}\epsilon^k
(2k+1)^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}+1\choose n_i-k} \equiv 0 \mod
(n_1+n_m+1){n_1+n_m\choose n_1}, {align*} and conjecture that for any
nonnegative integer $r$ and positive integer $s$ such that $r+s$ is odd, $$
\sum_{k=0}^{n}\epsilon ^k (2k+1)^{r}({2n\choose n-k}-{2n\choose n-k-1})^{s}
\equiv 0 \mod{{2n\choose n}}, $$ where $\epsilon=\pm 1$.
