#### Abstract

Here presented a generalization of Catalan numbers and Catalan triangles associated with two parameters based on the sequence characterization of Bell-type Riordan arrays. Among the generalized Catalan numbers, a class of large generalized Catalan numbers and a class of small generalized Catalan numbers are defined, which can be considered as an extension of large Schroder numbers and small Schroder numbers, respectively. Using the characterization sequences of Bell-type Riordan arrays, some properties and expressions including the Taylor expansions of generalized Catalan numbers are given. A few combinatorial interpretations of the generalized Catalan numbers are also provided. Finally, a generalized Motzkin numbers and Motzkin triangles are defined similarly. An interrelationship among parametrical Catalan triangle, Pascal triangle, and Motzkin triangle is presented based on the sequence characterization of Bell-type Riordan arrays.

#### Disciplines

Applied Mathematics | Mathematics

#### Recommended Citation

He, Tian-Xiao, "Parametric Catalan Numbers and Catalan Triangles" (2013). *Scholarship*. 63.

https://digitalcommons.iwu.edu/math_scholarship/63

## Comments

Linear Algebra and its Applicationsis published by Elsevier, http://www.journals.elsevier.com/linear-algebra-and-its-applications/.