How to prove this polynomial always has integer values at all integers

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.