On a Central Transform of Integer Sequences

TL;DR

This paper introduces a new transformation on integer sequences based on lower-triangular matrices, explores its properties including effects on Hankel transforms, and connects it to combinatorial sequences using Riordan array techniques.

Contribution

It defines a novel sequence transformation using matrix concepts and analyzes its properties, expanding the understanding of sequence transformations and their combinatorial applications.

Findings

Transformation affects Hankel transforms of sequences

Generates sequences with combinatorial significance

Utilizes Riordan array techniques for analysis

Abstract

We use the concept of the half of a lower-triangular matrix to define a transformation on integer sequences. We explore the properties of this transformation, including in some cases a study of the Hankel transform of the transformed sequences. Starting from simple sequences with elementary rational generating functions, we obtain many sequences of combinatorial significance. We make extensive use of techniques drawn from the theory of Riordan arrays.