Some relatives of the Catalan sequence

Abstract

We study a family of sequences $c_n(a_2,\dots ,a_r)$, where $r\ge 2$ and $a_2,\dots ,a_r$ are real parameters. We find a sufficient condition for positive definiteness of the sequence $c_n(a_2,\dots ,a_r)$ and check several examples from OEIS. We also study relations of these sequences with the free and monotonic convolution.