标题： 卷积序列，I：通过整数分拆函数的视角 显示英文标题

标题： Convolutive sequences, I: Through the lens of integer partition functions

摘要： 受具有指定和项的分拆计数函数卷积行为的启发，我们考虑满足通用卷积性质的原始eta-乘积的系数序列$(a_n)_{n\ge 0}$ \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end{align*} 对于特定的正整数$m$。 鉴于对在线整数序列百科全书中的此类序列进行彻底搜索的结果，范围为$m$到$6$，我们首先关注$m=2$的情况，主要关注两个$2$-卷积序列的组合学，分别为两者提供双射证明。 对于在OEIS中发现的其他$2$卷积序列，我们应用生成函数操作来证明它们的卷积性。 我们还给出了两个$3$卷积序列的例子。 最后，我们讨论了其他不是eta-乘积的卷积级数。 显示英文摘要

摘要： Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences $(a_n)_{n\ge 0}$ of primitive eta-products that satisfy the generic convolutive property \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end{align*} for a specific positive integer $m$. Given the results of an exhaustive search of the Online Encyclopedia of Integer Sequences for such sequences for $m$ up to $6$, we first focus on the case where $m=2$ with our attention mainly paid to the combinatorics of two $2$-convolutive sequences, featuring bijective proofs for both. For other $2$-convolutive sequences discovered in the OEIS, we apply generating function manipulations to show their convolutivity. We also give two examples of $3$-convolutive sequences. Finally, we discuss other convolutive series that are not eta-products.