Motzkin Numbers of Higher Rank: Generating Function and Explicit Expression

Toufik Mansour Department of Mathematics University of Haifa Haifa 31905 Israel

                toufik@math.haifa.ac.il

Matthias Schork^{1} Camillo-Sitte-Weg 25 60488 Frankfurt Germany

                mschork@member.ams.org

Yidong Sun Department of Mathematics Dalian Maritime University 116026 Dalian P. R. China

                 sydmath@yahoo.com.cn

Abstract

The generating function for the (colored) Motzkin numbers of higher rank introduced recently is discussed. Considering the special case of rank one yields the corresponding results for the conventional colored Motzkin numbers for which in addition a recursion relation is given. Some explicit expressions are given for the higher rank case in the first few instances.

1All correspondence should be directed to this author.

1 Introduction

The classical Motzkin numbers (\underline{A001006} in [1]) count the numbers of Motzkin paths (and are also related to many other combinatorial objects, see Stanley [2]). Let us recall the definition of Motzkin paths. We consider in the Cartesian plane ℤ × ℤ those lattice paths starting from (0, 0) that use the steps {U, L, D}, where U = (1, 1) is an up-step, L = (1, 0) a level-step and D = (1, −1) a down-step. Let M(n, k) denote the set of paths beginning in (0, 0) and ending in (n, k) that never go below the x-axis. Paths in M(n, 0) are called Motzkin paths and m_{n} := |M(n, 0)| is called n-th Motzkin number. Sulanke showed [3] that the Motzkin numbers satisfy the recursion relation

(n + 2)m_{n} = (2n + 1)m_{n−1} + 3(n − 1)m_{n−2}  (1)

and it is a classical fact (see Stanley [2]) that their generating function is given by

∑_{n≥0} m_{n}x^{n} = \frac{1 − x − \sqrt{1 − 2x − 3x^{2}}}{2x^{2}} .  (2)

Those Motzkin paths which have no level-steps are called Dyck paths and are enumerated by Catalan numbers (\underline{A000108} in [1]), see Stanley [2]. In recent times the above situation has been generalized by introducing colorings of the paths. For example, the k-colored Motzkin paths have horizontal steps colored by k colors (see [4, 5] and the references given therein). More generally, Woan introduced [6, 7] colors for each type of step. Let us denote by u the number of colors for an up-step U, by l the number of colors for a level-step L and by d the number of colors for a down-step D. (Note that if we normalize the weights as u + l + d = 1 we can view the paths as discrete random walks.) One can then introduce the set M^{(u,l,d)}(n, 0) of (u, l, d)-colored Motzkin paths and the corresponding (u, l, d)-Motzkin numbers m_{n}^{(u,l,d)} := |M^{(u,l,d)}(n, 0)|. Woan has given [6] a combinatorial proof that the (1, l, d)- Motzkin numbers satisfy the recursion relation

(n + 2)m_{n}^{(1,l,d)} = l(2n + 1)m_{n−1}^{(1,l,d)} + (4d − l^{2})(n − 1)m_{n−2}^{(1,l,d)}.  (3)

Choosing l = 1 and d = 1 yields the recursion relation (1) of the conventional Motzkin numbers m_{n} ≡ m_{n}^{(1,1,1)}. Note that choosing (u, l, d) = (1, k, 1) corresponds to the k-colored Motzkin paths. Defining m_{k,n} := |M^{(1,k,1)}(n, 0)|, one obtains from (3) the recursion relation (n + 2)m_{k,n} = k(2n + 1)m_{k,n−1} + (4 − k^{2})(n − 1)m_{k,n−2} for the number of k-colored Motzkin paths. Sapounakis and Tsikouras derived [4] the following generating function for m_{k,n}:

∑_{n≥0} m_{k,n}x^{n} = \frac{1 − kx − \sqrt{(1 − kx)^{2} − 4x^{2}}}{2x^{2}} .  (4)

For k = 1 this identity reduces to (2) for the conventional Motzkin numbers m_{n} ≡ m_{1,n}. One of the present authors suggested [8] (as “Problem 1”) that it would be interesting to find the recursion relation and generating function for the general (u, l, d)-Motzkin numbers m_{n}^{(u,l,d)}.

We will prove in Theorem 2.1 thet m_{n}^{(u,l,d)} = m_{n}^{(1,l,ud)}, yielding the desired recursion relation. Furthermore, a generating function and an explicit expression is derived for m_{n}^{(u,l,d)}. In [8] it was furthermore suggested to find the recursion relation and the generating function for the (colored) Motzkin numbers of higher rank (“Problem 2”). These numbers were introduced by Schork [9] in the context of “duality triads of higher rank” (and have also been considered before as “excursions”, see, e.g., [10]). In view of this connection Schork conjectured [8] that the Motzkin numbers of rank r satisfy a recursion relation of order 2r + 1. Very recently, Prodinger was the first to observe [11] that this conjecture does not hold by discussing explicitly the case r = 2. In particular, already for the first nontrivial cases r = 2, 3 the relations involved become very cumbersome. We will describe in Theorem 3.2 the generating function for the Motzkin numbers of higher rank and discuss then several particular cases explicitly.

2 Recursion relation and generating function for the general (colored) Motzkin numbers

Theorem 2.1. The general (u, l, d)-Motzkin numbers satisfy the recursion relation

(n + 2)m_{n}^{(u,l,d)} = l(2n + 1)m_{n−1}^{(u,l,d)} + (4ud − l^{2})(n − 1)m_{n−2}^{(u,l,d)}.  (5)

A generating function is given by

∑_{n≥0} m_{n}^{(u,l,d)}x^{n} = \frac{1 − lx − \sqrt{(1 − lx)^{2} − 4udx^{2}}}{2udx^{2}} ,  (6)

implying the explicit expression

m_{n}^{(u,l,d)} = ∑_{j=0}^{\frac{n}{2}} \frac{1}{j + 1}\binom{2j}{j}\binom{n}{2j}u^{j}d^{j}l^{n−2j}.  (7)

Proof. Let us prove that m_{n}^{(u,l,d)} = m_{n}^{(1,l,ud)} for all n ≥ 0. In order to see that, let U be any up-step in a Motzkin path; we call (U, D) a pair if the down-step D is the first down-step on the right-hand side of U which has the same height as U. The set of pairs of a Motzkin path is uniquely determined and for each pair exist ud possible combinations of colorings. The same number of possible colorings result if the up-steps u are always colored white (i.e., u = 1) and if each down-step can be colored by ud colors (which we call “alternative colors” to distinguish them from the original colors). Thus, given a (u, l, d)-Motzkin path we may replace the colors (u_{j}, d_{j}) (with 1 ≤ u_{j} ≤ u and 1 ≤ d_{j} ≤ d) for the j-th pair (U, D) by the combination of alternative colors (1, c_{j}) where c_{j} is the ([u_{j} − 1]d + d_{j})-th alternative color (with 1 ≤ c_{j} ≤ ud). Replacing the colors of all pairs in this fashion by the alternative colors yields a (1, l, ud)-Motzkin path. Thus, we have constructed a bijection between the set of (u, l, d)-Motzkin paths of length n and set of (1, l, ud)-Motzkin paths of length n, thereby showing that m_{n}^{(u,l,d)} = m_{n}^{(1,l,ud)}. From this identity and (3) we immediately obtain (5). An equation for the generating function M_{(u,l,d)}(x) := ∑_{n≥0} m_{n}^{(u,l,d)}x^{n} is obtained from the “first return decomposition” of a nonempty (u, l, d)-Motzkin path M: either M = LM′ or M = UM′DM′′, where M′, M′′ are (u, l, d)-Motzkin paths. The two possibilities give the contributions lxM_{(u,l,d)}(x) and udx^{2}(M_{(u,l,d)}(x))^{2}. Hence, M_{(u,l,d)}(x) satisfies

M_{(u,l,d)}(x) = 1 + lxM_{(u,l,d)}(x) + udx^{2}(M_{(u,l,d)}(x))^{2},

yielding

M_{(u,l,d)}(x) = \frac{1 − lx − \sqrt{(1 − lx)^{2} − 4udx^{2}}}{2udx^{2}}

which is the asserted equation (6) for the generating function. Note that this may also be written as

M_{(u,l,d)}(x) = \frac{1}{1 − lx}C (\frac{udx^{2}}{(1 − lx)^{2}} ) ,

where

C(y) = \frac{1 − \sqrt{1 − 4y}}{2y} = ∑_{n≥0} \frac{1}{n + 1}\binom{2n}{n}y^{n}

is the generating function for the Catalan numbers (see Stanley [2]). Thus,

M_{(u,l,d)}(x) = ∑_{j≥0} \frac{1}{j + 1}\binom{2j}{j}\frac{u^{j}d^{j}x^{2j}}{(1 − lx)^{2j+1}} .

Recalling M_{(u,l,d)}(x) = ∑_{n} m_{n}^{(u,l,d)}x^{n}, a comparison of coefficients shows that the number of (u, l, d)-Motzkin paths of length n is given by (7).

3 Generating function for the general (colored) Motzkin numbers of higher rank

We will now generalize the situation considered in the previous section to the case of higher rank. One of the present authors discussed [9] in the context of duality triads of higher rank (where one considers recursion relations of higher order, or equivalently, orthogonal matrix polynomials [12]) why it is interesting to consider the situation where the steps of the paths can go up or down more than one unit. The maximum number of units which a single step can go up or down will be called the rank. More precisely, let r ≥ 1 be a natural number. The set of admissable steps consists of:

1. r types of up-steps U_{j} = (1, j) with weights u_{j} for 1 ≤ j ≤ r.

2. A level-step L = (1, 0) with weight l.

3. r types of down-steps D_{j} = (1, −j) with weights d_{j} for 1 ≤ j ≤ r.

In the following we write (u, l, d) := (u_{r}, …, u_{1}, l, d_{1}, …, d_{r}) for the vector of weights.

Definition 3.1. [9] The set M^{(u,l,d)}(n, 0) of (u, l, d)-colored Motzkin paths of rank r and length n is the set of paths which start in (0, 0), end in (n, 0), have only admissable steps and are never below the x-axis. The corresponding number of paths, m_{n}^{(u,l,d)} := |M^{(u,l,d)}(n, 0)|, will be called (u, l, d)-Motzkin number of rank r.

Motzking paths of higher rank were considered in the literature already before [9] under the name “excursions”, see, e.g., [10]. In [8] it is discussed how one may associate with each Motzkin path of rank r and length n a conventional Motzkin path of length rn in a straightforward fashion. However, it was also discussed that this association is no bijection and that the case of higher rank is more subtle. In the following we derive an equation that the generating function for the Motzkin numbers of higher rank satisfies. In order to do that we need the following notations. We denote the set of all (u, l, d)-Motzkin paths of rank r that start at height s and end at height t (and have only admissable steps and are never below the x-axis) by A_{s,t}, and we denote the subset of paths of length n by A_{s,t}(n). Define

A_{s,t} ≡ A_{s,t}(x) := ∑_{n≥0} |A_{s,t}(n)|x^{n}.

We extend this notation by defining A_{s,t} = 0 for all s < 0 or t < 0. For s, t ≥ 0 we denote the subset of paths in A_{s,t} that never touch the x-axis by A_{s,t}^{∗}. In the following we make several times use of the relation A_{s,t}^{∗} ≃ A_{s−1,t−1}, or, |A_{s,t}^{∗}(n)| = |A_{s−1,t−1}(n)|, implying

A_{s,t}^{∗}(x) := ∑_{n≥0} |A_{s,t}^{∗}(n)|x^{n} = A_{s−1,t−1}(x).  (8)

Theorem 3.2. The generating function A_{0,0} ≡ M_{(u,l,d)}(x) = ∑_{n≥0} m_{n}^{(u,l,d)}x^{n} for the general (u, l, d)-Motzkin numbers of rank r satisfies

A_{0,0} = 1 + lxA_{0,0} + x^{2}A_{0,0} ∑_{p=1}^{r} ∑_{q=1}^{r} u_{p}d_{q}A_{p−1,q−1},  (9)

where the generating functions A_{i,j} with 1 ≤ i ≤ r − 1 and 0 ≤ j ≤ r − 1 satisfy

A_{i,j} = A_{i−1,j−1} + xA_{0,j} ∑_{q=1}^{r} d_{q}A_{i−1,q−1},

and for all 1 ≤ j ≤ r − 1,

A_{0,j} = xA_{0,0} ∑_{p=1}^{r} u_{p}A_{p−1,j−1}.

L  M′  ^{U_{p}  D_{q} M′  M′′}

Figure 1: First return decomposition of a Motzkin path of rank r.

Proof. From the definitions, a nonempty (u, l, d)-Motzkin path M can start either by a level step L or an up-step U_{p} with 1 ≤ p ≤ r. In the case the path starts by an up-step U_{p}, we use the “first return decomposition‘” of M, i.e., we write M as M = U_{p}M′D_{q}M′′ where M′′ is an arbitrary (u, l, d)-Motzkin path and M′ ∈ A_{p,q} such that M′ does not touch the height zero, see Figure 1. Thus, M′ ∈ A_{p,q}^{∗} and the generating function A_{0,0} satisfies

A_{0,0} = 1 + lxA_{0,0} + x^{2}A_{0,0} ∑_{p=1}^{r} ∑_{q=1}^{r} u_{p}d_{q}A_{p−1,q−1}

where we have used (8). This shows (9). Now, let us write an equation for the generating function A_{i,j} (with 1 ≤ i ≤ r−1 and 0 ≤ j ≤ r−1) for the number of (u, l, d)-Motzkin paths Q_{ij} = U_{i}P_{ij}D_{j} where P_{ij} ∈ A_{i,j}. In order to do that, we use the first return decomposition of such paths: either Q_{ij} = U_{i}P′_{ij}D_{j} such that P′_{ij} does not touch the height zero (i.e.,

U_{i}  P′_{ij}  D_{j}  ^{U_{i}  P′_{iq}  D_{q} P′′_{0j}  D_{j}}
Figure 2: The case 1 ≤ i ≤ r − 1 and 0 ≤ j ≤ r − 1.

P′_{ij} ∈ A_{i,j}^{∗}), or Q_{ij} = U_{i}P′_{iq}D_{q}P′′_{0j}D_{j} such that D_{q} is the first down-step that touches the height zero (thus, P′_{iq} does not touch the height zero, i.e., P′_{iq} ∈ A_{i,q}^{∗}) as described in Figure 2. Using (8), the generating function A_{i,j} thus satisfies

A_{i,j} = A_{i−1,j−1} + xA_{0,j} ∑_{q=1}^{r} d_{q}A_{i−1,q−1},

as claimed.

In order to write an equation for the generating function A_{0,j} (with 0 ≤ j ≤ r − 1), let us consider the last up-step U_{p} from height zero (there must exist at least one such up-step). In this case each (u, l, d)-Motzkin path can be decomposed as P′U_{p}P′′_{pj}D_{j}, where P′ is an

P′  U_{p} P′′_{pj}  D_{j}

Figure 3: The case i = 0 and 1 ≤ j ≤ r − 1.

arbitrary (u, l, d)-Motzkin path and P′′_{pj} ∈ A_{p,j} such that it does not touch the height zero (i.e., P′′_{pj} ∈ A_{p,j}^{∗}), as described in Figure 3. Thus, using again (8), the generating function A_{0,j} satisfies

A_{0,j} = xA_{0,0} ∑_{p=1}^{r} u_{p}A_{p−1,j−1},

which completes the proof.

Example 3.3. Let us consider as an example the case r = 1. It follows from Theorem 3.2 that

0 = 1 + (lx − 1)A_{0,0} + u_{1}d_{1}x^{2}A_{0,0}^{2},  (10)

yielding

A_{0,0} ≡ M_{(u_{1},l,d_{1})}(x) = \frac{1 − lx − \sqrt{(1 − lx)^{2} − 4u_{1}d_{1}x^{2}}}{2u_{1}d_{1}x^{2}}

as described in the proof of Theorem 2.1. This implies - according to (7) - that

m_{n}^{(u_{1},l,d_{1})} = ∑_{j=0}^{n/2} \frac{1}{1 + j} \binom{2j}{j}\binom{n}{2j}l^{n−2j}(u_{1}d_{1})^{j}.

Example 3.4. As another example, it follows from Theorem 3.2 that one has for r = 2 that

\{ \begin{matrix}A_{0,0} & = 1 + lxA_{0,0} + x^{2}A_{0,0}(u_{1}d_{1}A_{0,0} + u_{1}d_{2}A_{0,1} + u_{2}d_{1}A_{1,0} + u_{2}d_{2}A_{1,1}), \\ A_{0,1} & = xA_{0,0}(u_{1}A_{0,0} + u_{2}A_{1,0}), \\ A_{1,0} & = xA_{0,0}(d_{1}A_{0,0} + d_{2}A_{0,1}), \\ A_{1,1} & = A_{0,0} + xA_{0,1}(d_{1}A_{0,0} + d_{2}A_{0,1}). \end{matrix}

Solving the above system of equations we obtain that

0 = 1 + (lx − 1)A_{0,0} − x^{2}(d_{2}u_{2} − d_{1}u_{1})A_{0,0}^{2} + x^{2}(xu_{1}^{2}d_{2} + xu_{2}d_{1}^{2} − 2xlu_{2}d_{2} + 2d_{2}u_{2})A_{0,0}^{3}
−u_{2}d_{2}x^{4}(d_{2}u_{2} − d_{1}u_{1})A_{0,0}^{4} + x^{4}u_{2}^{2}d_{2}^{2}(lx − 1)A_{0,0}^{5} + x^{6}u_{2}^{3}d_{2}^{3}A_{0,0}^{6}.

Note that, if we set u_{2} = d_{2} = 0 in the above expression then we get (10). It is also interesting to consider the case (u, l, d) = (1, 1, 1) of non-colored Motzkin paths of rank 2, representing the most natural generalization of the conventional Motzkin paths. The above equation reduces in this case to

0 = 1 + (x − 1)A_{0,0} + 2x^{2}A_{0,0}^{3} + x^{4}(x − 1)A_{0,0}^{5} + x^{6}A_{0,0}^{6}
= (1 + xA_{0,0})^{2}(1 − (x + 1)A_{0,0} + x(x + 2)A_{0,0}^{2} − x^{2}(x + 1)A_{0,0}^{3} + x^{4}A_{0,0}^{4}).

Setting A_{0,0} ≡ M_{(1,1,1)}(x) and using that A_{0,0} is a formal power series (thus, A_{0,0} ≠ −\frac{1}{x}), this is equivalent to

0 = 1 − (x + 1)M_{(1,1,1)}(x) + x(x + 2)M_{(1,1,1)}^{2}(x) − x^{2}(x + 1)M_{(1,1,1)}^{3}(x) + x^{4}M_{(1,1,1)}^{4}(x), as described already in [10, 11]. To obtain a recursion relation for m_{n} ≡ m_{n}^{(1,1,1)}, one can use the MAPLE program package gfun written by Salvy et al. [13]. Prodinger has done this [11] to obtain

625(n + 3)(n + 2)(n + 1)m_{n} − 125(n + 3)(n + 2)(7n + 27)m_{n+1} − 50(n + 3)(5n^{2} + 24n + 23)m_{n+2} +(41890 + 30860n + 7540n^{2} + 610n^{3})m_{n+3} − (6844 + 5151n + 1214n^{2} + 91n^{3})m_{n+4}

−(n + 7)(23n^{2} + 301n + 976)m_{n+5} + 2(2n + 13)(n + 8)(n + 7)m_{n+6} = 0

and mentions that Salvy has informed him that this recursion of order 6 is minimal. Thus, the Motzkin numbers of rank r = 2 satisfy a 7-term recursion relation and not a 2·2+1 = 5term relation as the conjecture of Schork [8] implies. Thus, the conjecture does not hold! This was first observed by Prodinger [11]. The first few values of m_{n} ≡ m_{n}^{(1,1,1)} are given in Table 1; this sequence is sequence \underline{A104184} in [1].

n = 1 2 3 4 5 6  7  8  9  10
m_{n}^{(1,1,1)}  1 3 9 32 120 473 1925 8034 34188 147787

Table 1: The first few values of the non-colored Motzkin numbers of rank two.

The above example shows that the general case (where the weights are not restricted to 1) seems to be extremely complicated (see Theorem 3.2). Thus, from now on let us consider the case where all weights are equal to 1, i.e., u_{i} = l = d_{i} = 1 for 1 ≤ i ≤ r. In this case we denote the set of paths by B_{s,t} or B_{s,t}(n) and the generating function by B_{i,j} (instead of A_{i,j}). Then one has

B_{s,t} = B_{t,s}  (11)

which can be easily seen since to each path P ∈ B_{s,t}(n) one can associate a path P′ ∈ B_{t,s}(n) by traversing P in opposite direction. Since this is clearly a bijection the above equation follows. Theorem 3.2 shows that one has to solve in the general case a system of r^{2} equations for the r^{2} unknowns A_{i,j}. Due to the symmetry (11) this reduces in the case where all weights are equal to 1 to a system of \frac{r(r+1)}{2} equations in the \frac{r(r+1)}{2} unknowns B_{i,j} (where r − 1 ≥ i ≥ j ≥ 0).

Example 3.5. Theorem 3.2 with (11) gives for r = 3 the following set of \frac{3·4}{2} = 6 equations

\{ \begin{matrix}B_{0,0} & = 1 + xB_{0,0} + x^{2}B_{0,0}(B_{0,0} + 2B_{1,0} + 2B_{2,0} + 2B_{2,1} + B_{1,1} + B_{2,2}), \\ B_{1,0} & = xB_{0,0}(B_{0,0} + B_{1,0} + B_{2,0}), \\ B_{2,0} & = xB_{0,0}(B_{1,0} + B_{1,1} + B_{2,1}), \\ B_{1,1} & = B_{0,0} + xB_{1,0}(B_{0,0} + B_{1,0} + B_{2,0}), \\ B_{2,1} & = B_{1,0} + xB_{1,0}(B_{1,0} + B_{1,1} + B_{2,1}), \\ B_{2,2} & = B_{1,1} + xB_{2,0}(B_{1,0} + B_{1,1} + B_{2,1}). \end{matrix}

Solving the above system of equations we obtain that the generating function B_{0,0} ≡ M_{(1,1,1)}(x) for the non-colored Motzkin numbers of rank 3 satisfies

0 = 1 − (1 + x)B_{0,0} + 2xB_{0,0}^{2} + x^{2}(1 − 2x)B_{0,0}^{4} + 2x^{5}B_{0,0}^{6} − x^{6}(1 + x)B_{0,0}^{7} + x^{8}B_{0,0}^{8}.

Now, we would like to give a recursion relation for the sequence m_{n}^{(1,1,1)}. This can be automatically done with MAPLE’s program gfun: The procedure “algeqtodiffeq” translates the (algebraic) equation for B_{0,0} ≡ M_{(1,1,1)}(x) into an equivalent differential equation of order 7 (it is too long to present here) and then the procedure “diffeqtorec” translates the differential equation into a 28-term recursion relation. The first few values of m_{n}^{(1,1,1)} are given in Table 2; this sequence seems not to be listed in [1].

n = 1 2 3 4 5  6  7  8  9  10
m_{n}^{(1,1,1)}  1 4 16 78 404 2208 12492 72589 430569 2596471

Table 2: The first few values of the non-colored Motzkin numbers of rank three.

Example 3.6. Theorem 3.2 with (11) gives for r = 4 a set of \frac{4·5}{2} = 10 equations for the B_{i,j}. As above, it is possible to solve this system and obtain that the generating function B_{0,0} ≡ M_{(1,1,1)}(x) for the non-colored Motzkin numbers of rank 4 satisfies the following equation

0 = 1 + (x − 1)B_{0,0} − 2xB_{0,0}^{2} − x(x + 2)(x − 1)B_{0,0}^{3} − x^{2}(x − 2)(x + 2)B_{0,0}^{4}

+x^{2}(x − 1)B_{0,0}^{5} + x^{3}(x − 2)(x + 1)^{2}B_{0,0}^{6} + x^{4}(x + 1)(x − 1)^{2}B_{0,0}^{7} − x^{5}(2x^{2} − 3x − 4)B_{0,0}^{8} +x^{6}(x + 1)(x − 1)^{2}B_{0,0}^{9} + x^{7}(x − 2)(x + 1)^{2}B_{0,0}^{10} + x^{8}(x − 1)B_{0,0}^{11} − x^{10}(x − 2)(x + 2)B_{0,0}^{12}

−x^{11}(x + 2)(x − 1)B_{0,0}^{13} − 2x^{13}B_{0,0}^{14} + x^{14}(x − 1)B_{0,0}^{15} + x^{16}B_{0,0}^{16}.

The first few values of the corresponding Motzkin numbers m_{n}^{(1,1,1)} of rank 4 are given in Table 3; this sequence seems not to be listed in [1].

n = 1 2 3 4  5  6  7  8  9  10
m_{n}^{(1,1,1)}  1 5 25 155 1025 7167 51945 387000 2944860 22791189

Table 3: The first few values of the non-colored Motzkin numbers of rank four.

Remark. Let us consider the non-colored Motzkin numbers of rank r (i.e., all weights are equal to 1). They satisfy a τ(r)-term recursion relation where τ(r) is defined by this property (and is minimal). This yields a well-defined sequence {τ(r)}_{r∈ℕ} starting - according to the above examples - with 3, 7, 28, . . .. The original conjecture τ(r) = 2r + 1 proved to be too naive, but it seems that an exact formula for τ(r) will be very difficult to obtain. Thus, approximations or bounds would also be interesting. Considering the equations for the generating functions B_{0,0} ≡ M_{(1,1,1)}(x) of non-colored Motzkin numbers given in the above examples for rank r ≤ 4, one observes that they all have the form f(x, B_{0,0}) = 0 where f(x, y) is a polynomial of degree 2^{r} in y with coefficients in ℤ[x] (and with constant coefficient 1); more precisely, the coefficient a_{i}(x) of y^{i} is a polynomial of degree at most i in x over the integers ℤ. Thus, we can write

0 = 1 + ∑_{i=1}^{2^{r}} ∑_{j=0}^{i} a_{i,j}x^{j}(B_{0,0})^{i}  (12)

where a_{i,j} ∈ ℤ. It would be interesting to find out whether this representation holds for all ranks (which is what we expect) or whether it is confined to r ≤ 4.

We would like to close this paper by stressing that it is still an open problem to derive a recursion relation for the Motzkin numbers of higher rank. As the explicit examples r = 2, 3, 4 and the complicated set of equations for the generating function given in Theorem 3.2 show this will be a rather daunting task (see also the last remark).

4 Acknowledgments

The authors would like to thank Helmut Prodinger for sending them [11] prior to publication and drawing their attention to [10].

References

[1] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, published electronically at http://www.research.att.com/∼njas/sequences/, 1996—2007.

[2] R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999.

[3] R. Sulanke, Moments of generalized Motzkin paths, J. Integer Seq. 3 (2000), Article 00.1.1.

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2000 Mathematics Subject Classification: Primary, 05A15; Secondary, 11B37, 11B83. Keywords: Motzkin number; Catalan number; recursion relation; generating function.

(Concerned with sequences \underline{A001006}, \underline{A000108} and \underline{A104184}.)

Received April 20 2007; revised version received June 13 2007. Published in Journal of Integer Sequences, July 4 2007.

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