# On the halves of a Riordan array and their antecedents

## TL;DR

This paper explores the relationship between the horizontal and vertical halves of Riordan arrays, establishing conditions under which one half can be derived from the other and when a Riordan array is a half of another.

## Contribution

It unifies the study of the halves of Riordan arrays, providing new insights into their interrelation and criteria for when a Riordan array is a half of another.

## Key findings

- One half of a Riordan array can be obtained from the other under certain conditions.
- Criteria are established for when a Riordan array is a half of a larger Riordan array.
- The study connects the separate analyses of the array's halves into a unified framework.

## Abstract

Every Riordan array has what we call a horizontal half and a vertical half. These halves of a Riordan array have been studied separately before. Here, we place them in a common context, showing that one may be obtained from the other. We also ask and answer the question: given a Riordan array, when is it the half (either horizontal of vertical) of a Riordan array?