Several explicit and recursive formulas for generalized Motzkin numbers

Abstract

In the paper, the authors find two explicit formulas and recover a recursive formula for generalized Motzkin numbers. Consequently, the authors deduce two explicit formulas and a recursive formula for the Motzkin numbers, the Catalan numbers, and the restricted hexagonal numbers respectively.

Keywords

• explicit formula
• recursive formula
• generalized motzkin number
• motzkin number
• restricted hexagonal number
• catalan number
• generating function