Multinomial Catalan Numbers and Lucas Analogues

TL;DR

This paper introduces a new generalization of Catalan numbers using multinomial coefficients and explores their properties and integrality conditions, including Lucas analogues and regular Lucas sequences.

Contribution

It defines multinomial Catalan numbers and Lucas analogues, providing a complete characterization of when they are integers, expanding the understanding of these combinatorial sequences.

Findings

Characterization of integrality of multinomial Catalan numbers.

Complete analysis of Lucas analogue integrality conditions.

Identification of regular Lucas sequences with finitely many integer cases.

Abstract

We define a new generalization of Catalan numbers to multinomial coefficients. With arithmetic methods, we study their integrality and the integrality of their Lucasnomial generalization. We give a complete characterization of regular Lucas sequences for which they yield integers up to finitely many cases.