---
abstract: |
    The algebraic properties of formal power series with factorial growth which admit a certain well-behaved asymptotic expansion are discussed. It is shown that these series form a subring of ${\mathbb{R}}[[x]]$ which is also closed under composition of power series. An ‘asymptotic derivation’ is defined which maps a power series to its asymptotic expansion. Leibniz and chain rules for this derivation are deduced. With these rules asymptotic expansions of implicitly defined power series can be obtained. The full asymptotic expansions of the number of connected chord diagrams and the number of simple permutations are given as examples.

    **Keywords:** Asymptotic expansions; Formal power series; Chord diagrams; Simple permutations
author:
- 'Michael Borinsky[^1]'
bibliography:
- 'literature.bib'
title: |
    Generating Asymptotics\
    for factorially divergent sequences
---

Introduction
============

This article is concerned with sequences $a_n$, which admit an asymptotic expansion of the form, $$\begin{aligned}
 \label{eqn:first_example} a_n \sim \alpha^{n+\beta} \Gamma(n+\beta) \left( c_0 + \frac{c_1}{\alpha(n+\beta-1)} + \frac{c_2}{\alpha^2(n+\beta-1)(n+\beta-2)} + \ldots \right), \end{aligned}$$ for some $\alpha,\beta \in {\mathbb{R}}_{>0}$ and $c_k \in {\mathbb{R}}$. Sequences of this type appear in many enumeration problems, which deal with coefficients of factorial growth. For instance, generating functions of subclasses of permutations and graphs of fixed valence show this behaviour [@albert2003enumeration; 