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.
Highlights
The Motzkin numbers Mn enumerate various combinatorial objects
The Motzkin numbers Mn give the numbers of paths from (0, 0) to (n, 0) which never dip below the x-axis y = 0 and are made up only of the steps (1, 0), (1, 1), and (1, −1)
This implies that the generating function Ma,b(x) expressed in (5) is an explicit solution of the linear ordinary differential equations x2f (n)(x) + 2nxf (n−1)(x) + n(n − 1)f (n−2)(x) = Fn;a,b(x) for all n ≥ 2, where, by (19) and (20) or (21), n! 4b − a2 n (1 − ax)2 − 4bx2
Summary
Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China; College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China; Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China
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