Unimodality and log-concavity of generalized Glasby-Paseman sequences

TL;DR

This paper studies a two-parameter generalization of Glasby and Paseman sequences, conjecturing properties like unimodality and log-concavity, and proves these for specific parameters, with additional comments on the conjecture.

Contribution

It introduces a new two-parameter sequence, formulates conjectures about its properties, and proves these for the case where l=2 and a=1.

Findings

Conjectured unimodality and log-concavity based on computer experiments.

Proved properties for the case l=2, a=1.

Discussed asymptotic behavior and peak positions.

Abstract

In this paper, we consider a two-parameter ($l$ and $a$) generalization of a sequence that Glasby and Paseman considered. Based on computer experiments, we conjecture its unimodality, log-concavity, peak positions, and the asymptotic behavior of the maximum values. Then we prove this conjecture for the case where $l=2$ and $a=1$. We finish the paper by making some comments about the conjecture on the generalized sequence.