Using generating functions, we are proposing a unified approach to produce explicit formulas, which count the number of nodes in Smolyak grids based on various univariate quadrature or interpolation rules. Our approach yields, for instance, a new formula for the cardinality of a Smolyak grid, which is based on Chebyshev nodes of the first kind and it allows to recover certain counting-formulas previously found by Bungartz-Griebel, Kaarnioja, M\"uller-Gronbach, Novak-Ritter and Ullrich.

翻译：利用生成函数，我们提出了一种统一的方法来推导显式公式，用于计算基于不同单变量求积或插值规则的Smolyak网格中的节点数目。例如，我们的方法导出了基于第一类Chebyshev节点的Smolyak网格基数的新公式，并且能够恢复先前由Bungartz-Griebel、Kaarnioja、Müller-Gronbach、Novak-Ritter和Ullrich发现的某些计数公式。