Two explicit formulas for the generalized Motzkin numbers

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.


Introduction and main results
The Motzkin numbers M n enumerate various combinatorial objects. In , fourteen different manifestations of the Motzkin numbers M n were given in []. In particular, the Motzkin numbers M n give the numbers of paths from (, ) to (n, ) which never dip below the x-axis y =  and are made up only of the steps (, ), (, ), and (, -).
The first seven Motzkin numbers M n for  ≤ n ≤  are , , , , , , . All the Motzkin numbers M n can be generated by They can be connected with the Catalan numbers where x denotes the floor function whose value is the largest integer less than or equal to x. For detailed information, please refer to []    , 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  For k ≥ , the Motzkin numbers M k can be computed by where p q =  for q > p ≥  and the double factorial of negative odd integers -(n + ) is defined by Consequently, the Catalan numbers C k and the restricted hexagonal numbers H k can be computed by and respectively.
Theorem  For n ≥  and a, b ∈ N, the generalized Motzkin numbers M n (a, b) can be computed by

Consequently, equation (.) 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.

Proofs of main results
Now we are in a position to prove our main results.

Proof of Theorem  From (.), 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 denotes the falling factorial of x ∈ R. Consequently, by (.), it follows that for k ≥ , which can be rewritten as (.). The proof of Theorem  is complete.
Proof of Theorem  From (.), it is derived that This implies that By virtue of (.), (.), and (.), it follows that Substituting this into (.) and simplifying yield for k ≥ , which can be further rearranged as (.).

Proof of Theorem  For |x[(a  -b)x -a]| < , the generating function M a,b (x) in (.) can be expanded into
By (.) once again, it follows that , which means that

Remarks
Finally, we list several remarks.
Remark  The explicit formula (.) is a generalization of (.). , a, a  + b, a a  + b , a  + a  b + b  , a a  + a  b + b  , a  + a  b + a  b  + b  , a a  + a  b + a  b  + b  , a  + a  b + a  b  + a  b  + b  .

Remark  Equation (.) and many other alternative formulas for the Catalan numbers
In particular, the first nine restricted hexagonal numbers H n for  ≤ n ≤  are , , , , , , ,, ,, ,.

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.