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Two explicit formulas for the generalized Motzkin numbers
Journal of Inequalities and Applications volume 2017, Article number: 44 (2017)
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.
1 Introduction and main results
The Motzkin numbers \(M_{n}\) enumerate various combinatorial objects. In 1977, fourteen different manifestations of the Motzkin numbers \(M_{n}\) were given in [1]. In particular, the Motzkin numbers \(M_{n}\) 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 \((1,-1)\).
The first seven Motzkin numbers \(M_{n}\) for \(0\le n\le6\) are \(1, 1, 2, 4, 9, 21, 51\). All the Motzkin numbers \(M_{n}\) can be generated by
They can be connected with the Catalan numbers
by
where \(\lfloor x\rfloor\) denotes the floor function whose value is the largest integer less than or equal to x. For detailed information, please refer to [2] and the closely related references therein. For information on many results, applications, and generalizations of the Catalan numbers \(C_{n}\), please refer to the monographs [3, 4], the papers [5–13], the survey article [14], and the closely related references therein.
In [15], the \((u,l,d)\)-Motzkin numbers \(m_{n}^{(u,l,d)}\) were introduced and it was shown in [15], Theorem 2.1, that \(m_{n}^{(u,l,d)}=m_{n}^{(1,l,ud)}\),
and
Comparing (1.1) with (1.2) reveals that \(m_{n}^{(1,1,1)}=M_{n}\) and the \((u,l,d)\)-Motzkin numbers \(m_{n}^{(u,l,d)}\) generalize the Motzkin numbers \(M_{n}\).
In [16], the Motzkin numbers \(M_{n}\) were generalized in terms of the Catalan numbers \(C_{n}\) to
for \(a,b\in\mathbb{N}\) and the generating function
was discovered. It was pointed out in [2] that
where \(H_{n}\) denote the restricted hexagonal numbers and were described in [17].
For more information on many results, applications, and generalizations of the Motzkin numbers \(M_{n}\), please refer to [1, 2, 16, 18, 19] and the closely related references therein.
From (1.2) and (1.3), it is easy to see that \(m_{n}^{(u,l,d)}=m_{n}^{(d,l,u)}\). Comparing (1.2) with (1.4) reveals that \(M_{k}(a,b)\) and \(m_{k}^{(u,l,d)}\) are equivalent to each other and satisfy
Therefore, it suffices to consider the generalized Motzkin numbers \(M_{k}(a,b)\), rather than the \((u,l,d)\)-Motzkin numbers \(m_{n}^{(u,l,d)}\), in this paper.
The main aim of this paper is to establish explicit formulas for the Motzkin numbers \(M_{k}\) and the generalized Motzkin numbers \(M_{k}(a,b)\). As consequences, two explicit formulas for the restricted hexagonal numbers \(H_{n}\) are derived.
Our main results in this paper can be stated as the following theorems.
Theorem 1
For \(k\ge0\), the Motzkin numbers \(M_{k}\) can be computed by
where \(\binom{p}{q}=0\) for \(q>p\ge0\) and the double factorial of negative odd integers \(-(2n+1)\) is defined by
Theorem 2
For \(k\ge0\) and \(a,b\in\mathbb{N}\), the generalized Motzkin numbers \(M_{k}(a,b)\) can be computed by
Consequently, the Catalan numbers \(C_{k}\) and the restricted hexagonal numbers \(H_{k}\) can be computed by
and
respectively.
Theorem 3
For \(n\ge0\) and \(a,b\in\mathbb{N}\), the generalized Motzkin numbers \(M_{n}(a,b)\) can be computed by
Consequently, equation (1.9) for the Catalan numbers \(C_{n}\) is valid, the Motzkin numbers \(M_{n}\) and the restricted hexagonal numbers \(H_{k}\) can be computed by
and
respectively.
2 Proofs of main results
Now we are in a position to prove our main results.
Proof of Theorem 1
From (1.1), it follows that
This implies that
In combinatorial analysis, the Faà di Bruno formula plays an important role and can be described in terms of the Bell polynomials of the second kind
for \(n\ge k\ge0\), see [20], p.134, Theorem A, by
for \(n\ge0\); see [20], p.139, Theorem C. The Bell polynomials of the second kind \(\mathrm {B}_{n,k}(x_{1},x_{2}, \ldots, x_{n-k+1})\) satisfy the formula
for \(n\ge k\ge0\); see [20], p.135. In [21], Theorem 4.1, [10], Eq. (2.8), and [22], Section 3, it was established that
Then, for \(k\ge0\), we have
as \(x\to0\), where
denotes the falling factorial of \(x\in\mathbb{R}\). Consequently, by (2.1), it follows that
for \(k\ge0\), which can be rewritten as (1.7). The proof of Theorem 1 is complete. □
Proof of Theorem 2
From (1.4), it is derived that
This implies that
By virtue of (2.2), (2.3), and (2.4), it follows that
as \(x\to0\). Substituting this into (2.5) and simplifying yield
for \(k\ge0\), which can be further rearranged as (1.8).
Letting \((a,b)=(2,1)\) and \((a,b)=(3,1)\), respectively, in (1.8) and considering the last two relations in (1.5) lead to (1.9) and (1.10) immediately. The proof of Theorem 2 is complete. □
Proof of Theorem 3
For \(|x [ (a^{2}-4b )x-2a ] |<1\), the generating function \(M_{a,b}(x)\) in (1.4) can be expanded into
By (1.4) once again, it follows that
which means that
and
for \(n\in\mathbb{N}\). In conclusion, equation (1.11) follows.
Taking \((a,b)=(2,1)\), \((a,b)=(1,1)\), and \((a,b)=(3,1)\), respectively, in (1.11) and considering the three relations in (1.5) lead to (1.9), (1.12), and (1.13) readily. The proof of Theorem 3 is complete. □
3 Remarks
Finally, we list several remarks.
Remark 1
The explicit formula (1.8) is a generalization of (1.7).
Remark 2
Equation (1.9) and many other alternative formulas for the Catalan numbers \(C_{k}\) can also be found in [3–6, 8, 9, 12–14] and the closely related references therein.
Remark 3
By the second relation in (1.6), equation (1.3) can be reformulated as
which is different from the two equations (1.8) and (1.11).
Remark 4
Making use of any one among equations (1.8), (1.11), and (3.1), we can present the first nine generalized Motzkin numbers \(M_{n}(a,b)\) for \(0\le n\le8\) and \(a,b\in \mathbb{N}\) as follows:
In particular, the first nine restricted hexagonal numbers \(H_{n}\) for \(0\le n\le8\) are
4 Conclusions
By the Faà di Bruno formula and some properties of the Bell polynomials of the second kind, we establish two explicit formulas for the Motzkin numbers, the generalized Motzkin numbers, and the restricted hexagonal numbers.
References
Donaghey, R, Shapiro, LW: Motzkin numbers. J. Comb. Theory, Ser. A 23, 291-301 (1977)
Wang, Y, Zhang, Z-H: Combinatorics of generalized Motzkin numbers. J. Integer Seq. 18(2), 15.2.4 (2015)
Koshy, T: Catalan Numbers with Applications. Oxford University Press, Oxford (2009)
Stanley, RP: Catalan Numbers. Cambridge University Press, New York (2015). doi:10.1017/CBO9781139871495
Liu, F-F, Shi, X-T, Qi, F: A logarithmically completely monotonic function involving the gamma function and originating from the Catalan numbers and function. Glob. J. Math. Anal. 3, 140-144 (2015). doi:10.14419/gjma.v3i4.5187
Mahmoud, M, Qi, F: Three identities of the Catalan-Qi numbers. Mathematics 4(2), 35 (2016). doi:10.3390/math4020035
Qi, F, Guo, B-N: Logarithmically complete monotonicity of a function related to the Catalan-Qi function. Acta Univ. Sapientiae Math. 8, 93-102 (2016). doi:10.1515/ausm-2016-0006
Qi, F, Guo, B-N: Logarithmically complete monotonicity of Catalan-Qi function related to Catalan numbers. Cogent Math. 3, 1179379 (2016). doi:10.1080/23311835.2016.1179379
Qi, F, Mahmoud, M, Shi, X-T, Liu, F-F: Some properties of the Catalan-Qi function related to the Catalan numbers. SpringerPlus 5, 1126 (2016). doi:10.1186/s40064-016-2793-1
Qi, F, Shi, X-T, Liu, F-F, Kruchinin, DV: Several formulas for special values of the Bell polynomials of the second kind and applications. J. Appl. Anal. Comput. (2017, in press); (ResearchGate Technical Report (2015). Available online at doi:10.13140/RG.2.1.3230.1927)
Qi, F, Shi, X-T, Mahmoud, M, Liu, F-F: Schur-convexity of the Catalan-Qi function related to the Catalan numbers. Tbilisi Math. J. 9(2), 141–150 (2016). Available online at http://dx.doi.org/10.1515/tmj-2016-0026.
Qi, F, Shi, X-T, Mahmoud, M, Liu, F-F: The Catalan numbers: a generalization, an exponential representation, and some properties. J. Comput. Anal. Appl. 23, 937-944 (2017)
Shi, X-T, Liu, F-F, Qi, F: An integral representation of the Catalan numbers. Glob. J. Math. Anal. 3, 130-133 (2015). doi:10.14419/gjma.v3i3.5055
Qi, F: Some properties and generalizations of the Catalan, Fuss, and Fuss-Catalan numbers. ResearchGate Research (2015). Available online at doi:10.13140/RG.2.1.1778.3128
Mansour, T, Schork, M, Sun, Y: Motzkin numbers of higher rank: generating function and explicit expression. J. Integer Seq. 10, 07.7.4 (2007)
Sun, Z-W: Congruences involving generalized central trinomial coefficients. Sci. China Math. 57, 1375-1400 (2014). doi:10.1007/s11425-014-4809-z
Harary, F, Read, RC: The enumeration of tree-like polyhexes. Proc. Edinb. Math. Soc. (2) 17, 1-13 (1970)
Lengyel, T: Exact p-adic orders for differences of Motzkin numbers. Int. J. Number Theory 10, 653-667 (2014). doi:10.1142/S1793042113501157
Lengyel, T: On divisibility properties of some differences of Motzkin numbers. Ann. Math. Inform. 41, 121-136 (2013)
Comtet, L: Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged edn. Reidel, Dordrecht (1974)
Guo, B-N, Qi, F: Explicit formulas for special values of the Bell polynomials of the second kind and the Euler numbers. ResearchGate Technical Report (2015). Available online at doi:10.13140/2.1.3794.8808
Qi, F, Zheng, M-M: Explicit expressions for a family of the Bell polynomials and applications. Appl. Math. Comput. 258, 597-607 (2015). doi:10.1016/j.amc.2015.02.027
Acknowledgements
The first author was partially supported by China Postdoctoral Science Foundation with Grant Number 2015M582619.
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Zhao, JL., Qi, F. Two explicit formulas for the generalized Motzkin numbers. J Inequal Appl 2017, 44 (2017). https://doi.org/10.1186/s13660-017-1313-3
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DOI: https://doi.org/10.1186/s13660-017-1313-3