Recurrent Computation of Alternating Moments of Squared Binomial Coefficients

This paper addresses the computation of the alternating sum of squared binomial coefficients weighted by powers of integer indices. We describe an efficient exact method based on the derivatives of a finite hypergeometric polynomial and Stirling numbers of the second kind. A two-step recurrence relation for the derivatives at x = 1 is derived from the underlying differential equation. The proposed algorithm reduces the number of required arithmetic operations significantly compared to both direct summation and the previously known Kravchuk polynomial approach. Local benchmarking results demonstrate a substantial runtime advantage, achieving a speedup of up to 1062x in high-dimensional tests.