Factors of Alternating Convolution of the Gessel Numbers

TL;DR

This paper explores the relationship between Gessel numbers and super Catalan numbers, proving divisibility properties of their alternating convolutions using new summation techniques.

Contribution

It establishes a novel connection between Gessel and super Catalan numbers and proves a divisibility property of their alternating convolutions.

Findings

Gessel numbers relate closely to super Catalan numbers.

Alternating convolution of Gessel numbers is divisible by half of the super Catalan numbers.

New summation methods are introduced for the proof.

Abstract

The Gessel number $P(n,r)$ is the number of the paths in plane with $(1, 0)$ and $(0,1)$ steps from $(0,0)$ to $(n+r, n+r-1)$ that never touch any of the points from the set $\{(x,x)\in \mathbb{Z}^2: x\geq r\}$. We show that there is a close relationship between the Gessel numbers $P(n,r)$ and the super Catalan numbers $S(n,r)$. By using new sums, we prove that an alternating convolution of the Gessel numbers $P(n,r)$ is always divisible by \frac{1}{2}$S(n,r)$.