# Some relatives of the Catalan sequence

## TL;DR

This paper investigates a family of parameterized sequences related to the Catalan sequence, establishing conditions for their positive definiteness and exploring their connections with free and monotonic convolutions.

## Contribution

It introduces a new family of sequences, provides criteria for their positive definiteness, and links them to free and monotonic convolution theories.

## Key findings

- Derived sufficient conditions for positive definiteness.
- Identified connections with free convolution.
- Analyzed examples from OEIS.

## Abstract

We study a family of sequences $c_n(a_2,\ldots,a_r)$, where $r\ge2$ and $a_2,\ldots,a_r$ are real parameters. We find a sufficient condition for positive definiteness of the sequence $c_n(a_2,\ldots,a_r)$ and check several examples from OEIS. We also study relations of these sequences with the free and monotonic convolution.