Two integer sequences related to Catalan numbers

TL;DR

This paper proves a conjecture by Zeilberger that two related integer sequences derived from Catalan numbers are positive integers, revealing new properties of these sequences.

Contribution

It introduces and proves the integrality and positivity of sequences defined through a novel recursive relation involving Catalan numbers.

Findings

Sequences are positive integers

Recursive relations connect sequences to Catalan numbers

Confirms Zeilberger's conjecture

Abstract

We prove the following conjecture of Zeilberger. Denoting by $C_n$ the Catalan number, define inductively $A_n$ by $(-1)^{n-1}A_n=C_n+\sum_{j=1}^{n-1} (-1)^{j} \binom{2n-1}{2j-1} A_j \,C_{n-j}$ and $a_n=2A_n/C_n$. Then $a_n$ (hence $A_n$) is a positive integer.