A Polynomial Ring Connecting Central Binomial Coefficients and Gould's Sequence

TL;DR

This paper introduces a new polynomial ring structure that links central binomial coefficients and Gould's sequence, providing algebraic tools to analyze and compute these sequences and their transforms.

Contribution

It constructs a specialized multivariate polynomial quotient ring that connects two important combinatorial sequences and explores its algebraic properties and applications.

Findings

The ring structure encodes both sequences and their relationships.

Conditions for Gröbner basis formation are established.

Method for binomial transform calculation using the ring is demonstrated.

Abstract

We establish a novel connection between the central binomial coefficients $\binom{2n}{n}$ and Gould's sequence through the construction of a specialized multivariate polynomial quotient ring. Our ring structure is characterized by ideals generated from elements defined by polynomial recurrence relations, and we prove the conditions under which the set of polynomial generators forms a Gr\"obner basis. By exploring a specific variation of our ring structure, we demonstrate that expanding and evaluating polynomials within the ring yields both the central binomial coefficients and Gould's sequence. Additionally, we present a method for calculating the binomial transforms of these sequences using our ring's unique properties. This work provides new insights into the connections between two fundamental combinatorial sequences and introduces a new tool for integer sequence analysis, with potential applications in number theory and algebraic combinatorics.