Some Deep and Original Questions about the "critical exponents" of Generalized Ballot Sequences

TL;DR

This paper investigates the critical exponents of generalized ballot sequences, raises fundamental questions about their exact values and rationality, and acknowledges a significant recent answer by Michael Wallner.

Contribution

It introduces new questions about the critical exponents of generalized ballot sequences and highlights recent progress by Michael Wallner in solving these questions.

Findings

Questions about the exact values of critical exponents were answered by Michael Wallner.

The paper discusses the rationality of these critical exponents.

A donation was made to OEIS in honor of the progress.

Abstract

We numerically estimate the critical exponents of certain enumeration sequences that naturally generalize the famous Catalan and super-Catalan sequences, and raise deep and original questions about their exact values, and whether they are rational numbers. In this version we announce that our questions were brilliantly answered by Michael Wallner, and that the pledged donation to the OEIS, in his honor, was made.