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.
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Number Theory