The question is
How to prove this polynomial always has integer values at all integers?
$$P_m(x)=\sum_{i=0}^{m}\sum_{j=0}^{m}{x+j\choose j}{x-1\choose j}{j\choose i}{m\choose i}{i\choose m-j}\frac{3}{(2i-1)(2j+1)(2m-2i-1)}.$$
Our answer is supported by a Mathematica notebook (source and pdf).
September 2015
Wilberd van der Kallen