Going Back to Neil Sloane's FIRST LOVE (OEIS Sequence A435): On the Total Heights in Rooted Labeled Trees

TL;DR

This paper revisits Neil Sloane's first OEIS sequence, providing explicit formulas for moments of total heights in rooted labeled trees and aiming to find the limiting distribution.

Contribution

It offers rigorously computed moments for total heights in rooted labeled trees and commits to identifying the limiting distribution's probability density function.

Findings

Explicit formulas for the first twelve moments of total height

Normalized average total height of rooted labeled trees

A pledge to find the limiting distribution's density function

Abstract

In this tribute to Neil Sloane, we revisit the first sequence in the On-Line Encyclopedia of Integer Sequences, sequence A435 (1, 8, 78, 944, 13800, 237432, 4708144, 105822432, ...), that he encountered when he was a graduate student, and when normalized gives the average total height of rooted labeled trees. We state rigorously-computed explicit expressions for the first twelve moments of the random variable `total height' on rooted labeled trees, and pledge to donate to the OEIS 100 dollars in honor of the first to find an explicit expression for the probability density function of the limiting scaled probability distribution, as n goes to infinity.