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arXiv · 2206.03808

Factors of Alternating Convolution of the Gessel Numbers

Jovan Mikić
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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}$S(n,r)$.

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Jovan Mikić. 2022-06-08. Factors of Alternating Convolution of the Gessel Numbers. https://arxiv.org/abs/2206.03808

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