A study of a recursive sequence of polynomials revealing weighted Catalan Numbers

This paper examines the recursive sequence of polynomials $p_n(x)$, defined by $p_0(x)=x^2-2$ and $p_n(x)=p_{n-1}(x{)}^2-2$ for $n\ge 1$. It describes the field-theoretic motivations behind this sequence, derives a recursive formula for its coefficients, and identifies invariants that uncover combinatorial connections, including links to weighted Catalan numbers.