On a new congruence in the Catalan triangle

For 0 ≤ k ≤ 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}. The generalized Schröder number Schr(n,m,2) of order two represents the number of all lattice paths in the plane from the point (0,0) to the point (n,m), using steps (1,0), (0,1), and (1,1), that never go below the line y=2x. We present a new alternating convolution formula for the numbers C(2n,k) multiplied by a power of a binomial coefficient. 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 Schröder numbers of order two.