On a New Congruence in the Catalan Triangle

Abstract: For $0\leq k \leq n$, the number $C(n,k)$ represents the number of all lattice paths in the plane from the point $(0,0)$ to the point $(n,k)$, using steps $(1,0)$ and $(0,1)$, that never rise above the main diagonal $y=x$. The Fuss-Catalan number of order three $C^{(3)}_n$ represents the number of all lattice paths in the plane from the point $(0,0)$ to the point $(2n,n)$, using steps $(1,0)$ and $(0,1)$, that do not rise above the line $y=\frac{x}{2}$. We present a new alternating convolution formula for the numbers $C(2n,k)$. By using a new class of binomial sums that we call $M$ sums, we prove that this sum is divisible by $C^{(3)}_n$ and by the central binomial coefficient $\binom{2n}{n}$. We do this by examining the numbers $T(n,j)=\frac{1}{2n+1}\binom{2n+j}{j}\binom{2n+1}{n+j+1}$, for which we present a new combinatorial interpretation, connecting them to the generalized Schr\"{o}der numbers of order two.