Dennis EichhornAffiliation: University of California, Irvine Title Of Talk: Infinite Products that Enjoy a Curious Self-Convolutive Property Abstract: In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects called \emph{partitions with designated summands}. If we restrict our attention to $\mathrm{PDO}(n)$, the number of partitions with designated summands in which all parts are odd, a very curious property emerges. The very unexpected identity $$ \sum_{n=0}^\infty \mathrm{PDO}(2n)q^n = \left ( \sum_{n=0}^\infty \mathrm{PDO}(n)q^n \right )^2 $$ holds. That is, the sequence $\{\mathrm{PDO}(2n)\}_{n=0}^\infty$ is the convolution of the sequence $\{\mathrm{PDO}(n) \}_{n=0}^\infty$ with itself! We now refer to sequences with this property as ``$2$-convolutive.'' An exhaustive search of the over 385,000 entries in the Online Encyclopedia of Integer Sequences reveals only a very small handful of previously known $2$-convolutive sequences. In this talk, in joint work with Chern, Fu, and Sellers, we expand this short list of interesting $2$-convolutive sequences, and we use both combinatorial and analytic techniques to study $\mathrm{PDO}(n)$ and other partition functions that share this curious property. In addition, we begin the exploration of $3$-convolutive sequences (defined analogously), and we present several unsolved problems in this area.
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Last update made Tue Mar 10 21:28:03 CDT 2026.
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