Several explicit and recursive formulas for generalized Motzkin numbers

Feng Qi; Bai-Ni Guo; Feng Qi; Bai-Ni Guo

doi:10.3934/math.2020091

AIMS Mathematics

2020, Volume 5, Issue 2: 1333-1345. doi: 10.3934/math.2020091

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Research article

Several explicit and recursive formulas for generalized Motzkin numbers

Feng Qi ^1,2,3,
Bai-Ni Guo ^{3
,
,}

1.
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China
2.
School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China
3.
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454010, China

Received: 06 November 2019 Accepted: 09 January 2020 Published: 20 January 2020
MSC : Primary: 05A15; Secondary: 05A19, 05A20, 11B37, 11B83, 34A05

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.
- explicit formula,
- recursive formula,
- generalized Motzkin number,
- Motzkin number,
- restricted hexagonal number,
- Catalan number,
- generating function
Citation: Feng Qi, Bai-Ni Guo. Several explicit and recursive formulas for generalized Motzkin numbers[J]. AIMS Mathematics, 2020, 5(2): 1333-1345. doi: 10.3934/math.2020091

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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.

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