Arithmetic Terms for Multinomial Coefficient Sums
Joseph M. Shunia, Lorenzo Sauras-Altuzarra
TL;DR
The paper develops a digit-extraction framework to express key combinatorial sums—partial sums of binomial coefficients, their multisections, and polynomial coefficients—as explicit arithmetic terms. It introduces and leverages the digit-extraction theorem to recover coefficients from polynomial evaluations, enabling Kronecker-style encodings and constructive arithmetic-term formulas. The authors derive closed-form arithmetic-term expressions for partial sums, multisection sums, and multinomial-type polynomial coefficients, and apply these to central trinomial coefficients, notably giving a concrete arithmetic-term representation for $[x^n](x^2+x+1)^n$ via $\binom{n,3}{n} = \left\lfloor \left(\frac{3^{3n}-1}{3^{2n}-3^n}\right)^n \right\rfloor \bmod 3^n$. This yields a definitive, explicit arithmetic-term solution related to a longstanding problem of Graham et al., highlighting the practical reach of arithmetic terms in combinatorics and number theory.
Abstract
We construct arithmetic terms representing the partial sums of binomial coefficients, and we extend these results to obtain arithmetic terms representing the multisections of binomial coefficient sums. We also introduce an arithmetic term representing a certain type of multinomial coefficient sum and, as an application, we provide an arithmetic term representing the central trinomial coefficients. This solves one of the research problems of the celebrated book Concrete Mathematics, which remained open for nearly thirty years.