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

TL;DR

This paper explores a recursive polynomial sequence linked to weighted Catalan numbers, revealing algebraic and combinatorial properties through recursive formulas and invariants.

Contribution

It introduces a new recursive polynomial sequence and uncovers its connections to weighted Catalan numbers with field-theoretic insights.

Findings

Derived recursive formulas for polynomial coefficients

Identified invariants linking to weighted Catalan numbers

Revealed combinatorial structures underlying the sequence

Abstract

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 $n \geq 1$. 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.