Several explicit and recursive formulas for generalized Motzkin numbers

: 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.


Introduction
The Motzkin numbers M n enumerate various combinatorial objects.In 1977, Donaghey and Shapiro [3] gave fourteen different manifestations of the Motzkin numbers M n .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 ≤ n ≤ 6 are 1, 1, 2, 4, 9, 21, 51.All the Motzkin numbers M n can be generated by In 2007, Mansour et al [12] introduced the (u, l, d)-Motzkin numbers m (u,l,d) From (1.1) and (1.2), it is easy to see that m (u,l,d) n = m (d,l,u) n .In 2014, Sun [42] generalized the Motzkin numbers M n to for a, b ∈ N in terms of the Catalan numbers and established the generating function where λ denotes the floor function defined by the largest integer less than or equal to λ ∈ R. Wang and Zhang pointed out in [43] that where H n denote the restricted hexagonal numbers described by Harary and Read [4].For more information on many results, applications, and generalizations of the Motzkin numbers M n , please refer to the papers [3,9,10,42,43] and closely related references therein.For more information on many results, applications, and generalizations of the Catalan numbers C n , please refer to the monograph [5], the newly published papers [11, 17, 19, 26, 27, 31, 36-38, 40, 41], the survey articles [25,29], and closely related references therein.
Comparing (1.1) with (1.5) reveals that M k (a, b) and m (u,l,d)   k are equivalent to each other and satisfy (1.9) Consequently, the generating function M a,b (x) defined by (1.5) in [42] is defined for either b ≤ 0 or a ≥ 2 √ b > 0. In this paper, we will find two explicit formulas, which are different from (1.8), and recover the recursive formula (1.9) for generalized Motzkin numbers M n (a, b).Consequently, we will derive two explicit formula and a recursive formula for the Motzkin numbers M n , the Catalan numbers C n , and the restricted hexagonal numbers H n respectively.
We can state our main results as the following three theorems.
Theorem 1.For n ≥ 0, we can compute generalized Motzkin numbers M n (a, b) by where p q = 0 for q > p ≥ 0 and the double factorial of negative odd integers −(2n + 1) is Consequently, we can compute the Motzkin numbers M n and the restricted hexagonal numbers H n respectively by and Consequently, we can compute the Motzkin numbers M n and the restricted hexagonal numbers H n respectively by and the recursive formula (1.9).Consequently, for n ≥ 0, the Motzkin numbers M n , the Catalan numbers C n , and the restricted hexagonal numbers H n meet the recursive formulas and respectively.

Lemmas
In order to prove the explicit formula (1.10), we need the following lemmas.).Let u(x) and v(x) 0 be two differentiable functions.Let U (n+1)×1 (x) be an Then the nth derivative of the ratio u(x) v(x) can be computed by Lemma 2 ( [2, p. 134, Theorem A and p. 139, Theorem C]).The Faà di Bruno formula can be described in terms of the Bell polynomials of the second kind for n ≥ 0.

Proofs of Theorems 1 and 3
We are now in a position to prove our main results.
Proof of Theorem 1.By virtue of (2.1), (2.2), and (2.3), we obtain for k ≥ 0 that as x → 0, where Therefore, by L'Hôspital's rule, we have making use of (3.1), and simplifying lead to the explicit formula ( respectively in (1.10) and considering the three relations in (1.6) derive (1.11) and (1.12) immediately.The proof of Theorem 1 is complete.
Proof of Theorem 2. From (1.5), it is derived that This implies that It is easy to see that 1. when a 2 − 4b > 0 and x ≤ min By virtue of (3.2), we obtain the formula (1.13) readily.Letting (a, b) = (1, 1) and (a, b) = (3, 1) respectively in (1.13) and making use of the first and third relations in (1.6) lead to (1.11) and (1.12) immediately.The proof of Theorem 2 is complete.
Proof of Theorem 3. From (1.5), it is derived that Squaring on both sides of the above equation gives .
Remark 3.This paper is a continuation of the article [49] and a revised version of the preprint [28].

. 7 )
Therefore, it suffices to consider generalized Motzkin numbers M k (a, b), rather than the (u, l, d)-Motzkin numbers m (u,l,d) n , in this paper.By the second relation in (1.7), one can reformulated the formula (1.2) as M n (a, b) = a n into (1.3)recovers (1.8) once again.In 2015, Wang and Zhang [43, Theorem 1] combinatorially obtained, among other things, the recursive formula