Some results and conjectures about Hankel determinants of sequences which are related to Catalan-like numbers

TL;DR

This paper explores properties of Hankel determinants related to Catalan-like numbers, extending previous work by examining other columns and shifts through theoretical analysis and computer experiments.

Contribution

It introduces new properties of Hankel determinants for columns beyond the first and considers backward shifts via matrix modifications.

Findings

Properties of Hankel determinants for other columns are identified.

Computer experiments suggest new conjectures about these determinants.

Backward shifts of sequences are analyzed through matrix modifications.

Abstract

Martin Aigner introduced Catalan-like numbers as elements of the first column of admissible matrices and studied Hankel determinants of their forward shifts. In this paper we collect some properties of the Hankel determinants of the other columns which are suggested by computer experiments. By prepending zero rows to admissible matrices we also consider Hankel determinants of backward shifts.