title: Two explicit formulas for the generalized Motzkin numbers.
abstract: In the paper, by the Faà di Bruno formula, the authors establish two explicit formulas for the Motzkin numbers, the generalized Motzkin numbers, and the restricted hexagonal numbers.
authors.affiliation: State Key Laboratory of Integrated Service Networks, Xidian University, Xi'an, Shaanxi 710071 China.|Department of Mathematics and Physics, Weinan Normal University, Weinan, Shaanxi 714009 China.
authors.forename: Jiao-Lian
authors.initials: JL
authors.lastname: Zhao
authors.affiliation: Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin, 300387 China.|Institute of Mathematics, Henan Polytechnic University, Jiaozuo, Henan 454010 China.
authors.forename: Feng
authors.identifier: 0000-0001-6239-2968
authors.initials: F
authors.lastname: Qi
keywords: Bell polynomial of the second kind; Catalan number; Faà di Bruno formula; Motzkin number; explicit formula; generalized Motzkin number; generating function; restricted hexagonal number