Sums of powers of Catalan triangle numbers

In this paper, we consider combinatorial numbers ${\left(C_{m,k}\right)}_{m\ge 1,k\ge 0}$, mentioned as Catalan triangle numbers where $C_{m,k}?\left(\genfrac{}{}{0ex}{}{m?1}{k}\right)?\left(\genfrac{}{}{0ex}{}{m?1}{k?1}\right)$. These numbers unify 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, 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 $C_{2n,n?1}=C_{2n+1,n}=C_n$.We present identities for sums (and alternating sums) of $C_{m,k}$, squares and cubes of $C_{m,k}$ and, consequently, for $B_{n,k}$ and $A_{n,k}$. In particular, one of these identities solves an open problem posed in Gutiérrez et al. (2008). We also give some identities between ${\left(C_{m,k}\right)}_{m\ge 1,k\ge 0}$ and harmonic numbers ${\left(H_n\right)}_{n\ge 1}$. Finally, in the last section, new open problems and identities involving ${\left(C_n\right)}_{n\ge 0}$ are conjectured.