Continuants, run lengths, and Barry's modified Pascal triangle

TL;DR

This paper reveals a novel connection between Barry's modified Pascal triangle and continuants of binary run lengths, providing explicit formulas for diagonal and row sums.

Contribution

It introduces a new interpretation of Barry's modified Pascal triangle using continuants of binary run lengths, with explicit formulas for sums.

Findings

Diagonal sums correspond to continuants of binary run lengths.

Explicit formulas for row sums are derived.

The connection offers new combinatorial insights.

Abstract

We show that the $n$'th diagonal sum of Barry's modified Pascal triangle can be described as the continuant of the run lengths of the binary representation of $n$. We also obtain an explicit description for the row sums.