On one property of Catalan numbers

TL;DR

This paper presents a new proof demonstrating a specific divisibility property of Catalan numbers, showing that for certain odd indices not congruent to 1 mod 3, the number is divisible by n+2.

Contribution

It provides a novel proof of a divisibility property of Catalan numbers, expanding understanding of their arithmetic characteristics.

Findings

Catalan number C_n divisible by n+2 under specified conditions

New proof technique for Catalan number properties

Enhanced understanding of divisibility patterns in Catalan numbers

Abstract

We give a new proof of the following statement: the Catalan number $C_n$ is divisible by $n+2$, if $n$ is odd and $n\not\equiv 1\text{ mod }3$.