Sums of powers of Catalan triangle numbers

TL;DR

This paper introduces new identities and sum formulas for combinatorial numbers unifying Catalan triangle entries, including sums of powers, recurrence relations, and connections to harmonic numbers, solving an open problem and proposing future conjectures.

Contribution

It presents novel identities and sum formulas for the unified Catalan triangle numbers, including solutions to open problems and new conjectures.

Findings

Derived new recurrence relations and sum identities for $C_{m,k}$

Solved an open problem related to sums of powers of Catalan triangle numbers

Proposed new conjectures involving Catalan numbers and harmonic numbers

Abstract

In this paper we consider combinatorial numbers $C_{m, k}$ for $m\ge 1$ and $k\ge 0$ 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.