A study of a recursive sequence of polynomials revealing weighted Catalan Numbers
Abstract
This paper examines the recursive sequence of polynomials pn(x), defined by p0(x) = x2 - 2 and pn(x) = pn-1(x)2 - 2 for n ≥ 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.
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