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

TL;DR

This paper proves that a specific complex polynomial, defined by a double sum of binomial coefficients and rational functions, always takes integer values at all integer inputs, addressing a problem posed on Mathoverflow.

Contribution

The paper provides a rigorous proof that the given polynomial is integer-valued at all integers, resolving an open problem from Mathoverflow.

Findings

The polynomial is integer-valued at all integers.

The proof involves combinatorial and algebraic techniques.

The result confirms the conjecture posed by Kevin.

Abstract

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.