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Two explicit formulas for the generalized Motzkin numbers
Jiao-Lian Zhao1,2, Feng Qi3,4
1State Key Laboratory of Integrated Service Networks, Xidian University, Xi'an, Shaanxi 710071 China.
Journal of Inequalities and Applications
|March 4, 2017
Summary
This paper introduces two new explicit formulas for calculating Motzkin numbers, generalized Motzkin numbers, and restricted hexagonal numbers. These formulas utilize the Faà di Bruno formula for precise mathematical derivations.
Area of Science:
- Combinatorics and Number Theory
- Discrete Mathematics
Background:
- Motzkin numbers and their generalizations appear in various counting problems.
- Restricted hexagonal numbers have applications in tiling and lattice path counting.
- The Faà di Bruno formula provides a method for expressing derivatives of composite functions.
Purpose of the Study:
- To derive novel explicit formulas for specific combinatorial sequences.
- To extend the application of the Faà di Bruno formula in number theory.
- To provide efficient computational methods for Motzkin and hexagonal numbers.
Main Methods:
- Application of the Faà di Bruno formula.
- Algebraic manipulation and combinatorial identities.
- Derivation of explicit recurrence relations.
Main Results:
- Two distinct explicit formulas for Motzkin numbers were established.
- Generalized Motzkin numbers were shown to be expressible via the derived formulas.
- Explicit formulas for restricted hexagonal numbers were successfully derived.
Conclusions:
- The Faà di Bruno formula offers a powerful tool for enumerative combinatorics.
- The derived formulas provide new insights into the structure of Motzkin and hexagonal numbers.
- This work contributes to the computational and theoretical understanding of these number sequences.
Keywords:
Bell polynomial of the second kindCatalan numberFaà di Bruno formulaMotzkin numberexplicit formulageneralized Motzkin numbergenerating functionrestricted hexagonal numberMore Related Videos
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