Convolutions of Tribonacci, Fuss-Catalan, and Motzkin sequences

TL;DR

This paper introduces a new class of sequences defined via Bell polynomials, encompassing well-known sequences like Catalan and Motzkin, and provides a general convolution formula with illustrative examples.

Contribution

It defines a broad class of sequences using partial Bell polynomials, unifying several known sequences and deriving a general convolution formula.

Findings

Derived a general multifold convolution formula for Bell sequences

Included explicit examples demonstrating the convolution formula

Unified various well-known sequences within a single framework

Abstract

We introduce a class of sequences, defined by means of partial Bell polynomials, that contains a basis for the space of linear recurrence sequences with constant coefficients as well as other well-known sequences like Catalan and Motzkin. For the family of `Bell sequences' considered in this paper, we give a general multifold convolution formula and illustrate our result with a few explicit examples.