A matrix with sums of Catalan numbers -- LU-decomposition and determinant

TL;DR

This paper investigates a matrix constructed from sums of neighboring Catalan numbers, providing its LU-decomposition and revealing that its determinant equals an odd Fibonacci number, thus confirming previous combinatorial findings.

Contribution

The paper presents the LU-decomposition of a Catalan number-based matrix and proves its determinant equals an odd Fibonacci number, linking combinatorial sequences with matrix theory.

Findings

LU-decomposition of the matrix is explicitly given.

Determinant of the matrix equals an odd Fibonacci number.

Confirmation of earlier combinatorial results through algebraic methods.

Abstract

Following Benjamin et al., a matrix with entries being sums of two neighbouring Catalan numbers is considered. Its LU-decomposition is given, by guessing the results and later prove it by computer algebra, with lots of human help. Specializing a parameter, the determinant turns out to be a Fibonacci number with odd index, confirming earlier results, obtained back then by combinatorial methods.