Sums of powers of Catalan triangle numbers

In this paper, we consider combinatorial numbers (Cm,k)m >= i,k >= 0, mentioned as Catalan triangle numbers where C-m,C-k := ((m-1)(k)) - ((m-1) (k-1)). These numbers unify the entries of the Catalan triangles B-n,B-k and A(n,k) for appropriate values of parameters in and k, i.e., B-n,B-k = C-2n,C-n-k and A(n,k) = C-2n+1,C-n+1-k. In fact, these numbers are suitable rearrangements of the known ballot numbers and some of these numbers are the well-known Catalan numbers C-n, that is C-2n,C-n-1 = C-2n+1,C-n = C-n. We present identities for sums (and alternating sums) of C-m,C-k squares and cubes of C-m,C-k and, consequently, for B-n,B-k and A(n,k). In particular, one of these identities solves an open problem posed in Gutierrez et al. (2008). We also give some identities between (C-m,C-k)(m >= 1,1k >= 0) and harmonic numbers (H-n)n >= 1. Finally, in the last section, new open problems and identities involving (C-n)(n >= 0) are conjectured. (C) 2017 Elsevier B.V. All rights reserved.