Some new binomial sums related to the Catalan triangle (Q405119)

Summary: In this paper, we derive many new identities on the classical Catalan triangle \(\mathcal{C}=(C_{n,k})_{n\geq k\geq 0}\), where \(C_{n,k}=\frac{k+1}{n+1}\binom{2n-k}{n}\) are the well-known ballot numbers. The first three types are based on the determinant and the fourth is relied on the permanent of a square matrix. It not only produces many known and new identities involving Catalan numbers, but also provides a new viewpoint on combinatorial triangles.