2025/07/15 by Shane Chern, Dennis Eichhorn, Chern, Shane +5 · 1 citation
Mathematics · #05A17 #11P81 #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2507.10965
openalex publication_date 2025/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences (an)n≥ 0 of primitive eta-products that satisfy the generic convolutive property ∑n≥ 0 amn qn = (∑n≥ 0 an qn)m 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.