Recently a new generalization of Pascal's triangle, the family of so-called hyperbolic Pascal triangles (in short $\mathcal{HPT}$) was introduced. In this paper, we consider a variation of $\mathcal{HPT}$ by replacing the two leg-sequences of the triangle by arbitrary sequences ${\{{\alpha }_n\}}_{n\ge 0}$ and ${\{{\beta }_n\}}_{n\ge 0}$. Originally the legs were the constant 1 sequence. We describe some quantitative properties via recurrences relations, explicit formulae and generating functions. In addition, we present a useful general result on linear recurrences sequences diverted by an arbitrary sequence.

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双曲帕斯卡三角形的推广

最近帕斯卡三角形的新推广，即所谓的双曲帕斯卡三角形家族（简称 $\mathcal{高压电}$) 的介绍。在本文中，我们考虑一种变体$\mathcal{高压电}$ 通过用任意序列替换三角形的两条腿序列 ${\{{\alpha }_n\}}_{n\ge 0}$ 和 ${\{{\beta }_n\}}_{n\ge 0}$. 最初的腿是常数 1 序列。我们通过递推关系、显式公式和生成函数来描述一些定量属性。此外，我们提出了关于由任意序列转向的线性递归序列的有用的一般结果。