Convolutions of Liouvillian sequences

Abstract

While Liouvillian sequences are closed under many operations, simple examples show that they are not closed under convolution, and the same goes for d'Alembertian sequences. Nevertheless, we show that d'Alembertian sequences are closed under convolution with rationally d'Alembertian sequences, and that Liouvillian sequences are closed under convolution with rationally Liouvillian sequences.

Some general results in combinatorial enumeration

“…P k (n) are the partitions of n with k parts, and similarly Q k (n) are those with k distinct parts. The sequence p : N → N is[154, A000041] and begins (p(n)) n≥1 =(1,2,3,5,7,11,15,22,30,42,56,77,101,135, 176, . .…”