Catalan-like Numbers and Determinants

✍ Scribed by Martin Aigner

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 190 KB
• Volume: 87
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1998.2945

✦ Synopsis

A class of numbers, called Catalan-like numbers, are introduced which unify many well-known counting coefficients, such as the Catalan numbers, the Motzkin numbers, the middle binomial coefficients, the hexagonal numbers, and many more. Generating functions, recursions and determinants of Hankel matrices are computed, and some interpretations are given as to what these numbers count.

each have determinant 1, for all n. It was shown in [1] that, for the Motzkin numbers, the determinant of the first Hankel matrix is again 1 for all n, while the determinant of the second matrix is 1, 0, &1, &1, 0, 1 for n=1, ..., 6, repeating modulo 6 thereafter.