Sums of powers of Catalan triangle numbers

2016/02/13 by Miana, Pedro J., Ohtsuka, Hideyuki, Romero, Natalia
#05A10 #05A19 #11B65 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1602.04347

Abstract

In this paper we consider combinatorial numbers C_{m, k} for m≥ 1 and k≥ 0 which unifies the entries of the Catalan triangles B_{n, k} and A_{n, k} for appropriate values of parameters m and k, i.e., B_{n, k}=C_2n,n-k and A_{n, k}=C_2n+1,n+1-k. In fact, some of these numbers are the well-known Catalan numbers C_n that is C_2n,n-1=C_2n+1,n=C_n. We present new identities for recurrence relations, linear sums and alternating sum of C_m,k. After that, we check sums (and alternating sums) of squares and cubes of C_m,k and, consequently, for B_{n, k} and A_{n, k}. In particular, one of these equalities solves an open problem posed in \cite[GHMN]. We also present some linear identities involving harmonic numbers H_n and Catalan triangles numbers C_m,k. Finally, in the last section new open problems and identities involving C_n are conjectured.

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