# Periodic oscillations of coefficients of power series that satisfy functional equations, a practical revision

## TL;DR

This paper derives complete asymptotics for power series coefficients of solutions to certain functional equations and applies these results to improve asymptotic estimates for counting specific combinatorial structures.

## Contribution

It provides a comprehensive asymptotic analysis of coefficients for solutions to a class of functional equations and enhances existing combinatorial enumeration results.

## Key findings

- Derived complete asymptotics for power series coefficients.
- Improved asymptotic estimate for the number of 2,3-trees with n leaves.
- Methods applicable to more general functional equations.

## Abstract

For the solutions $\Phi(z)$ of functional equations $\Phi(z)=P(z)+\Phi(Q(z))$, we derive a complete asymptotic of power series coefficients. As an application, we improve significantly an asymptotic of the number of $2,3$-trees with $n$ leaves given in Adv. Math. 44:180-205, 1982 by Andrew M. Odlyzko. The methods we consider can be applied to more general functional equations too.