Catalan三角数的幂和
Sums of powers of Catalan triangle numbers
摘要：
在本文中，我们考虑组合数$C_{m, k}$为$m\ge 1$和$k\ge 0$，它统一了加泰罗尼亚三角形$ B_{n, k}$和$ A_{n, k}$的条目，参数$m$和$k$的适当值，即，$B_{n, k}=C_{2n,n-k}$和$A_{n, k}=C_{2n+1,n+1-k}$。事实上，其中一些数字是著名的加泰罗尼亚数字$C_n$，即$C_{2n,n-1}=C_{2n+1,n}=C_n$。给出了$C_{m,k}$的递推关系、线性和及交错和的新恒等式.之后，我们检查$C_{m,k}$的平方和和立方的和（以及交替和），从而检查$ B_{n, k}$和$ A_{n, k}$。特别是，这些等式之一解决了\cite{[GHMN]}中提出的一个开放问题。我们还提出了一些涉及调和数$H_n$和Catalan三角形数$C_{m,k}$的线性恒等式.最后，在最后一节中，新的开放问题和涉及$C_n$的身份被澄清。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
