Factors of Alternating Convolution of the Gessel Numbers

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 $\{(x,x)\in {\mathbb{Z}}^2:x\ge r\}$. 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}$S(n,r)$.