The $O(1/n^{85})$ Asymptotic expansion of OEIS sequence A85

TL;DR

This paper derives a highly precise asymptotic expansion of OEIS sequence A85, which counts involutions, improving the approximation from O(1/n) to O(1/n^{85}), showcasing advanced asymptotic analysis.

Contribution

The paper presents the first derivation of an O(1/n^{85}) asymptotic expansion for the involution sequence, significantly refining previous approximations.

Findings

Derived the O(1/n^{85}) asymptotic formula

Enhanced understanding of involution sequence behavior

Connects combinatorics with advanced asymptotic techniques

Abstract

One of the most important sequences in enumerative combinatorics is OEIS sequence A85, the number of involutions of length n. In the Art of Computer Programming, vol. 3, Don Knuth derived the O(1/n) asymptotic formula for these numbers. In this modest tribute to our two heroes, Neil Sloane who just turned 85, and Don Knuth who was 85 a year ago, we go all the way to an $O(1/n^{85})$ asymptotic formula.