Proofs of two conjectures on Catalan triangle numbers

TL;DR

This paper proves two conjectures related to sums of products of Catalan triangle numbers, employing Zeilberger's algorithm and q-analogues to establish the results, thus advancing combinatorial number theory.

Contribution

It introduces novel proofs for two conjectures on Catalan triangle numbers, one using Zeilberger's algorithm and the other through q-analogues.

Findings

Proved the first conjecture using Zeilberger's algorithm.

Established the second conjecture via q-analogues.

Confirmed the conjectures originally posed by Miana, Ohtsuka, and Romero.

Abstract

We prove two conjectures on sums of products of Catalan triangle numbers, which were originally conjectured by Miana, Ohtsuka, and Romero [Discrete Math. 340 (2017), 2388--2397]. The first one is proved by using Zeilberger's algorithm, and the second one is proved by establishing its $q$-analogue.