  The following problem was posed by user "Kevin" on Mathoverflow. How to prove
this polynomial always has integer values at all integers?
$$P_m(x)=\sum_{i=0}^{m}\sum_{j=0}^{m} \binom{x+j}{j} \binom{x-1}{j}
\binom{j}{i} \binom{m}{i} \binom{i}{m-j} \frac{3}{(2i-1)(2j+1)(2m-2i-1)}.$$
  We provide an answer.
