A note on number triangles that are almost their own production matrix

TL;DR

This paper explores a special class of number triangles with production matrices closely linked to the triangles themselves, revealing combinatorial significance and connections to convolution recurrences and continued fractions.

Contribution

It characterizes a family of number triangles with nearly self-related production matrices and investigates their combinatorial and algebraic properties.

Findings

Identified a family of number triangles with specific production matrix relations.

Connected certain triangles to convolution recurrences and continued fraction generating functions.

Provided insights into the combinatorial significance of these triangles.

Abstract

We characterize a family of number triangles whose production matrices are closely related to the original number triangle. We study a number of such triangles that are of combinatorial significance. For a specific subfamily, these triangles relate to sequences that have interesting convolution recurrences and continued fraction generating functions.