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Polynomial with squared binomial coefficients

Source: IMAR Test 2023 P4

December 16, 2023
polynomialalgebra

Problem Statement

Let nn{} be a non-negative integer and consider the standard power expansion of the following polynomial ∑k=0n(nk)2(X+1)2k(X−1)2(n−k)=∑k=02nakXk.\sum_{k=0}^n\binom{n}{k}^2(X+1)^{2k}(X-1)^{2(n-k)}=\sum_{k=0}^{2n}a_kX^k.The coefficients a2k+1a_{2k+1} all vanish since the polynomial is invariant under the change X↦−X.X\mapsto -X. Prove that the coefficients a2ka_{2k} are all positive.
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