On iterated generating functions for integer sequences, and catalan polynomials

Abstract

We explore the notion of linearly convergent iterated generating functions - families of finite polynomials containing strings of coefficients which converge towards an infinite sequence. Examples of naturally occurring generating algorithms are given, and we show how some of these can be recovered by imposing a recurrence scheme on an individual sequence and applying term matching (to solve for unknown constants). Computer automation of hand procedures results in the appearance of so called Catalan polynomials whose role in generating finite Catalan subsequences we identify and then formalise as a theorem.