Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers

2020/11/21 by Paul Barry, Barry, Paul
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2011.10827

Abstract

We provide a context around a conjectured closed form for the Hankel transform of linear combinations of consecutive pairs of Catalan numbers. This generalizes the formula for the Hankel transforms of the shifted Catalan numbers and the known results for linear combinations of pairs of Catalan numbers. Many interesting number triangles emerge, some of which we analyze using the language of Riordan arrays.

Citations

• Production Matrices and Riordan Arrays
• The on-line encyclopedia of integer sequences

Related

• The Hankel Transform of the Sum of Consecutive Generalized Catalan Numbers
• Shifted Hankel determinants of Catalan numbers and related results II: Backward shifts
• Conjectures on Somos 4, 6 and 8 sequences using Riordan arrays and the Catalan numbers
• Some remarks about Hankel determinants which are related to Catalan-like numbers
• Simple closed form Hankel transforms based on the central coefficients of certain Pascal-like triangles