A Generalization of the K-bonacci Sequence from Riordan Arrays

In this article, we introduce a family of weighted lattice paths, whose step set Using these lattice paths, we define a family of Riordan arrays whose sum on the rising diagonal is the k-bonacci sequence. This construction generalizes the Pascal and Delannoy Riordan arrays, whose sum on the rising diagonal is the Fibonacci and tribonacci sequence , respectively. From this family of Riordan arrays we introduce a generalized k-bonacci polynomial sequence, and we give a lattice path combinatorial interpretation of these polynomials. In particular, we find a combinatorial interpretation of tribonacci and tribonacci-Lucas polynomials.


Introduction
A lattice path Γ in the xy-plane with steps in a given set S ⊂ Z 2 is a concatenation of directed steps of S, i.e., Γ = s 1 s 2 • • • s l , where s i ∈ S for 1 i l.For any positive integer n, let M 1 (n, k) denote the set of lattice paths from the point (0, 0) to the point (k, n), with step set S 1 = {H = (1, 0), V = (0, 1)}.Let D 1 (n, k) be the number of lattice paths of M 1 (n, k), i.e., D 1 (n, k) = |M 1 (n, k)|.It is well known that Let H 1 be the infinite lower triangular array defined by , where Therefore, H 1 is the Pascal matrix.This matrix has a lot of interesting properties, (see for instance [2,5,6,7,12,23,24]), one of them is the sum of the elements on the rising diagonal is the Fibonacci sequence F n , (sequence A000045) 1 .
If we add a third step, D 1 = (1, 1), we obtain Delannoy paths; among other references see [1,9,10,13,22].We denote by M 2 (n, k) the set of lattice paths from the point (0, 0) to the point (k, n), with step set S 2 = {H = (1, 0), V = (0, 1), The sequence D 2 (n, n) is called central Delannoy numbers (sequence A001850).Let H 2 be the infinite lower triangular array defined by , where This array is called Delannoy or tribonacci matrix.It satisfies that the sum of the elements on the rising diagonal is the tribonacci sequence t n (sequence A000073).
A generalized Delannoy number D * 2 (n, k) is the number of weighted lattice paths such that the steps H, V and D are labelled with weights a, b and c, respectively.This kind of paths is called (a, b, c)-weighted paths [9].The generalized Delannoy numbers are given by the following formula (cf.[15]): Cheon et al. [9] defined the generalized Delannoy triangle H 2 (a, b, c) = [d n,k ] n,k∈N , where and, they studied this array by means of Riordan arrays.The first few terms of this triangle are Additionally, they introduced a generalized Lucas polynomial sequence, and they gave a lattice path combinatorial interpretation for these polynomials.In particular, they found a combinatorial interpretation of the polynomials of Fibonacci, Lucas, Pell, Pell-Lucas Chebyschev of first and second kind, among others.The study of lattice paths through Riordan arrays is not new, see [18,21].
A natural questions is: What family of steps are required to find a Riordan array, such that the sum of the elements on its rising diagonal is the k-bonacci numbers F (k) n ?The k-bonacci numbers are defined by the recurrence In particular, we obtain the Fibonacci sequence and tribonacci sequence, in the case k = 2, 3, respectively.
In this paper, we introduce a family of weighted lattice paths whose step set is where each step is labelled with weights a 1 , a 2 , . . ., a m+1 , respectively.From the number of these weighted lattice paths, we obtain a new family of Riordan arrays, such that the sum of the elements on its rising diagonal is the k-bonacci sequence.Moreover, we obtain a generalized k-bonacci polynomial sequence.The new family of weighted lattice paths leads us to a combinatorial interpretation for the generalized k-bonacci polynomial sequence.In particular, we study a generalization of the tribonacci polynomials and tribonacci-Lucas polynomials.We also show three combinatorial interpretations of the central tetrabonacci numbers.

Riordan arrays
the electronic journal of combinatorics 22(1) (2015), #P1.38 The first column being indexed by 0. From the definition, follows where [z n ] is the coefficient operator.The matrix corresponding to the pair f (z), g(z) is denoted by R(g(z), f (z)) or (g(z), f (z)).The product of two Riordan arrays (g(z), f (z)) and (h(z), l(z)) is defined by: The set of all Riordan matrices is a group under the operator * , (cf.[19]).The identity element is I = (1, z), and the inverse of (g(z), Example 1.The Pascal matrix H 1 and the Delannoy matrix H 2 are given by the following Riordan arrays, respectively.
) The generalized Delannoy array H 2 (a, b, c) has a Riordan array expression given by The following theorem is known as the fundamental theorem of Riordan arrays or summation property (cf.[21]).
where h(z) is the generating function of the sequence {h k } k∈N .

Generalized Tribonacci and Tribonacci-Lucas polynomial sequence
In this section, we defined a new kind of lattice paths: (a, b, c, d)-weighted paths.From the number of (a, b, c, d)-weighted lattice paths, we introduce a new infinite lower triangular array.Then, we study its properties and we obtain the generalized tribonacci and the electronic journal of combinatorics 22(1) (2015), #P1.38 tribonacci-Lucas polynomial sequence.
Let M 3 (n, k) denote the set of (a, b, c, d)-weighted paths from the point (0, 0) to the point (k, n), with step set S 3 = {H = (1, 0), V = (0, 1), D 1 = (1, 1), D 2 = (1, 2)}, where each step is labelled with weights a, b, c and d, respectively.The weight of a weighted path is the product of the weights of all its steps in the weighted path and the length of a weighted path is the number of steps making up the path.For example, in Figure 1, we show a (a, b, c, d)-lattice path of length 5 and weight a 2 bcd.
Theorem 4. The number of (a, b, c, d)-lattice paths is given by Proof.For n 1, let Then by Equation (2), we obtain the electronic journal of combinatorics 22(1) (2015), #P1.38 Then, Therefore, by the binomial theorem , where The first few terms of this triangle array are The infinite triangular array H 3 (a, b, c, d) has a Riordan array expression given by Proof.The proof runs like in Theorem 4, because the electronic journal of combinatorics 22(1) (2015), #P1.38 Proposition 7. Let A 3 (z) be the generating function for the rows sums of the Riordan array H 3 .Then (3) Proof.By applying Theorem 3 to the Riordan array H 3 with the generating function Let F n (x) be the n-th row sum of H 3 , then from above proposition we obtain the generating function for The polynomials F n (x) are called generalized tribonacci polynomials.The first few terms are 1, p, p 2 + q, p 3 + 2pq + r, p 4 + 3p 2 q + 2pr + q 2 , p 5 + 4p 3 q + 3p 2 r + 3pq 2 + 2qr, . . ., where q = q(x), p = p(x), r = r(x).If p = x 2 , q = x and r = 1, we get the tribonacci polynomials (cf.[16]).
Theorem 8.For all integer n 0, and for any polynomials p(x), q(x), r(x) Proof.The generating function of the sum of elements on the rising diagonal lines (1diagonal) in the generalized Delannoy triangle H 2 (q(x), p(x), r(x)) is Then, by Equation ( 1) the electronic journal of combinatorics 22(1) (2015), #P1.38 We give an alternative proof of Theorem 8. Observe the equality Therefore by Theorem 3, polynomials F n (x) can be expressed by n-th sum of the rows of the following Riordan array , where q = q(x), p = p(x), r = r(x).So, the identity is clear.Following the same ideas of Theorem 8 in [17], we obtain the following identity: Proposition 9.For all integer n 0, and for any polynomials p(x), q(x), r(x) Theorem 10.For all integer n 3, the polynomials F n (x) satisfy the following recurrence: n−1 (x) + q(x)F n−2 (x) + r(x)F (3) where F (3) 2 (x) = p 2 (x) + q(x).Proof.The result follows from the equality We can write the polynomials F (3) n (x) as a Binet-like formula, i.e., , where α(x), β(x) and γ(x) are the roots of the characteristic equation of (5).
Proposition 12.The generating function of the tribonacci-Lucas polynomials is Proof.From Equation ( 6), we have the following matrix equation: where A T is the transpose of the matrix A.
Then by applying Theorem 3 to the Riordan array R(1 + q(x)z 2 + 2r(x)z 3 , z) with the generating function A 3 (z), we get the generating function: Theorem 13.Let p(x) = ax + b, q(x) = cx, and r(x) = dx.Then the electronic journal of combinatorics 22(1) (2015), #P1.38 Proof.By applying Theorem 3 to the Riordan array H 3 with the generating function h(z) = 1/(1 − xz), we have The coefficients of the above polynomial are the numbers of (a, b, c, d)-lattice paths.

Combinatorial Interpretations
The following theorem shows a combinatorial interpretation for the generalized tribonacci and tribonacci-Lucas polynomial sequences.Theorem 14.For any polynomial F where ω n,k (x) := ω n,k = h Proof.By definition, F n (x) is the n-th row sum of the Riordan array H 3 .Then (i) follows from Definition 5. Identity (ii) follows from Equation ( 6).
From Theorem 6 of [3], we have that the generating function of the central sequence d On the other hand, a Motzkin path of length n is a lattice path of Z × Z running from (0, 0) to (n, 0) that never passes below the x-axis and whose permitted steps are the up diagonal step U = (1, 1), the down diagonal step D = (1, −1) and the horizontal step H = (1, 0), called rise, fall and level step, respectively.The number of Motzkin paths of length n is the n-th Motzkin number m n , (sequence A001006).Many other examples of bijections between Motzkin numbers and others combinatorial objects can be found in [4].A Grand Motzkin path of length n is a Motzkin path without the condition that never passes below the x-axis.The number of Grand Motzkin paths of length n is the n-th Grand Motzkin number g n , sequence A002426.A a H b U -Motzkin path is a Motzkin path such that each horizontal step is colored with one of a specific colors and each up diagonal step is colored with one of b specific colors.The number of a Then, we get the following corollary.

Riordan Arrays and Generalized k-bonacci numbers
In this section, we generalize the results of the above section.We introduce a new family of weighted lattice paths, the A s -weighted paths.
Let A s = (a 1 , a 2 , . . ., a s ) be a vector of weights.Then, we denote by M m (n, k) the set of A m+1 -weighted paths from the point (0, 0) to the point (k, n), with step set with k m − 1, n 1 and initial conditions D m (0, k) = a k 2 and D m (n, 0) = a n 1 .
Theorem 19.The number of A m+1 -lattice paths is given by the electronic journal of combinatorics 22(1) (2015), #P1.38 Proof.For n 1, let Then from Equation ( 9), we have Then, Therefore, by the binomial theorem , where the electronic journal of combinatorics Proof.It is clear from Theorem 19.
For example, the first few terms of the Riordan array H 4 (a, b, c, d, e) are Let A m (z) be the generating function for the rows sums of the Riordan array H m .Then Proof.The proof runs like in Proposition 7.
Let F n (x) be the n-th row sum of H k , then from above proposition we obtain the generating function for where p 1 (x) = a 1 + a 2 , and p j (x) = a j+1 , j = 2, . . ., k.The polynomials F (k) n (x) are called generalized k-bonacci polynomials.
For example, if k = 4 we obtain the tetrabonacci polynomials.The first few terms of the tetrabonacci polynomials sequence are The sum on the rising diagonal in the Pascal triangle and in the tribonacci triangle are the Fibonacci numbers and tribonacci numbers, respectively.In general, we have the following proposition.
Proposition 23.The k-bonacci triangle T k defined by the following Riordan array satisfies that the sum on the rising diagonal is the k-bonacci sequence F From Theorem 3 and Theorem 19, we have the following identity.