SEVERAL EXPLICIT AND RECURSIVE FORMULAS FOR THE GENERALIZED MOTZKIN NUMBERS

In the paper, the authors find two explicit formulas and recover a recursive formula for the 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 [10] introduced the (u, l, d)-Motzkin numbers m (u,l,d) n and obtained [10,Theorem 2.1] and From ( 1) and ( 2), it is easy to see that m .In 2014, Sun [21] 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 [22] that where H n denote the restricted hexagonal numbers described by Harary and Read [5].
For more information on many results, applications, and generalizations of the Motzkin numbers M n , please refer to the papers [3,7,8,21,22] 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 [6], the papers [9,14,15,20], the survey article [12], and closely related references therein.
Comparing (1) with ( 5) reveals that M k (a, b) and m 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 (u,l,d) n , in this paper.By the second relation in (7), one can reformulated the formula (2) as Substituting ( 4) into (3) recovers (8) In this paper, we will find two explicit formulas, different from (8), and recover the recursive formula (9) for the 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 the generalized Motzkin numbers M n (a, b) by where p q = 0 for q > p ≥ 0 and the double factorial of negative odd integers Consequently, we can compute the Motzkin numbers M n and the restricted hexagonal numbers H n respectively by and (2 Theorem 2. For n ≥ 0, we can compute the generalized Motzkin numbers M n (a, b) by Consequently, we can compute the Motzkin numbers M n and the restricted hexagonal numbers H n respectively by and and the recursive formula (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 (10), we need the following lemmas.
for 1 ≤ i ≤ n + 1 and 1 ≤ j ≤ n, and let |W (n+1)×(n+1) (x)| denote the determinant of the (n + 1) × (n + 1) matrix 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.

Preprints
More generally, for n ≥ k ≥ 0 and λ ∈ R, we have 3. Proofs of Theorems 1 and 3 We are now in a position to prove our main results.
Proof of Theorem 2. From (5), it is derived that This implies that (1) when a 2 − 4b ≥ 0 and x ≤ min we have By virtue of ( 23), we obtain the formula (13) readily.Letting (a, b) = (1, 1) and (a, b) = (3, 1) respectively in (13) and making use of the first and third relations in (6) lead to (11) and ( 12) immediately.The proof of Theorem 2 is complete.
Proof of Theorem 3. From (5), it is derived that Squaring on both sides of the above equation gives  9) and considering the three relations in (6) lead to (15), (16), and ( 17) immediately.The proof of Theorem 3 is complete.

Two remarks
Remark 1. From the proof of Theorem 1, we can conclude that This implies that the generating function M a,b (x) expressed in ( 5) is an explicit solution of the linear ordinary differential equations x 2 f (n) (x) + 2nxf (n−1) (x) + n(n − 1)f (n−2) (x) = F n;a,b (x) for all n ≥ 2, where, by (19) and (20) or (21), .
Remark 2. This paper is a company and continuation of the article [24].