In this paper, we present two explicit expressions of the moments $M_{2k}$ of the Kesten distribution in terms of Catalan's triangles $T_{k,i}=\frac{k-i+1}{k+1}\left(\begin{array}{c}k+i\\ k\end{array}\right)$ and Shapiro's Catalan triangles $B_{k,j}=\frac{j}{k}\left(\begin{array}{c}2k\\ k-j\end{array}\right)$, respectively. As a corollary, we provide a generalization of the identity, due to Eplett, between the Catalan number $C_k=\frac{1}{k+1}\left(\begin{array}{c}2k\\ k\end{array}\right)$ and the triangles $B_{k,j}$. As an application, we obtain an asymptotic formula on $M_{2k}$ as k tends to infinity.

中文翻译：

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关于 Kesten 分布矩的注释

在本文中，我们提出了矩的两个显式表达式 ${米}_{2克}$ 以加泰罗尼亚三角形表示的 Kesten 分布 ${吨}_{克,\mathrm{一世}}=\frac{克-\mathrm{一世}+1}{克+1}\left(\begin{array}{c}克+\mathrm{一世}\\ 克\end{array}\right)$ 和夏皮罗的加泰罗尼亚三角形 ${乙}_{克,j}=\frac{j}{克}\left(\begin{array}{c}2克\\ 克-j\end{array}\right)$， 分别。作为推论，由于 Eplett，我们提供了加泰罗尼亚数之间的恒等式的概括$C_{克}=\frac{1}{克+1}\left(\begin{array}{c}2克\\ 克\end{array}\right)$ 和三角形 ${乙}_{克,j}$. 作为应用，我们得到了一个渐近公式${米}_{2克}$因为k趋于无穷大。