Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   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 ll-known Catalan numbers $C_n$ that is $C_{2n,n-1}=C_{2n+1,n}=C_n$. #R##N# present new identities for recurrence relations, linear sums and alternating sum of $C_{m,k}$. After that,  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]}.  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.
['Pedro J. Miana', 'Hideyuki Ohtsuka', 'Natalia Romero']
I am writing a research paper on a idea Sums of powers of Catalan triangle numbers
[{"from": "human", "value": "I am writing a research paper on a idea Sums of powers of Catalan triangle numbers"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   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 ll-known Catalan numbers $C_n$ that is $C_{2n,n-1}=C_{2n+1,n}=C_n$. #R##N# present new identities for recurrence relations, linear sums and alternating sum of $C_{m,k}$. After that,  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]}.  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."}]