  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.
