Sums of powers of Catalan triangle numbers

Abstract

In this paper, we consider combinatorial numbers (Cm,k)m≥1,k≥0, mentioned as Catalan triangle numbers where Cm,k≔−1k−m−1k−1. These numbers unify the entries of the Catalan triangles Bn,k and An,k for appropriate values of parameters m and k, i.e., Bn,k=C2n,n−k and An,k=C2n+1,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 Cn that is C2n,n−1=C2n+1,n=Cn. We present identities for sums (and alternating sums) of Cm,k, squares and cubes of Cm,k and, consequently, for Bn,k and An,k. In particular, one of these identities solves an open problem posed in Gutiérrez et al. (2008). We also give some identities between (Cm,k)m≥1,k≥0 and harmonic numbers (Hn)n≥1. Finally, in the last section, new open problems and identities involving (Cn)n≥0 are conjectured. © 2017 Elsevier B.V.