Factors of alternating convolution of the Gessel numbers

The Gessel number $P(n,r)$ is the number of lattice paths in the 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 Gessel numbers $P(n,r)$ and super Catalan numbers $T(n,r)$. A new class of binomial sums, so called $M$ sums, is used. By using one form of the Pfaff–Saalschütz theorem, a new recurrence relation for $M$ sums is proved. Finally, we prove that an alternating convolution of Gessel numbers $P(n,r)$ multiplied by a power of a binomial coefficient is always divisible by $\frac{1}{2}T(n,r)$.