A Generalization of the k-Bonacci Sequence from Riordan Arrays Jos´eL. Ram´ırez ∗ V´ıctor F. Sirvent Departamento de Matem´aticas Departamento de Matem´aticas Universidad Sergio Arboleda Universidad Sim´on Bol´ıvar Bogot´a, Colombia Caracas, Venezuela [email protected] [email protected] Submitted: Aug 21, 2014; Accepted: Jan 15, 2015; Published: Feb 16, 2015 Mathematics Subject Classifications: 11B39, 05A19, 15A24. Abstract In this article, we introduce a family of weighted lattice paths, whose step set is H = (1, 0),V = (0, 1),D = (1, 1),...,Dm = (1, m 1) . Using these lat- { 1 −1 − } tice 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 se- quence, respectively. From this family of Riordan arrays we introduce a generalized k-bonacci polynomial sequence, and we give a lattice path combinatorial interpre- tation of these polynomials. In particular, we find a combinatorial interpretation of tribonacci and tribonacci-Lucas polynomials. Keywords: Riordan arrays, k-bonacci sequence, lattice paths. 1 Introduction A lattice path Γ in the xy-plane with steps in a given set S Z2 is a concatenation of ⊂ directed steps of S, i.e., Γ = s s sl, where si S for 1 6 i 6 l. For any positive 1 2 ··· ∈ integer n, let M1(n, k) denote the set of lattice paths from the point (0, 0) to the point (k,n), with step set S1 = H = (1, 0),V = (0, 1) . Let 1(n, k) be the number of lattice paths of M (n, k), i.e., ({n, k)= M (n, k) . It is} well knownD that 1 D1 | 1 | n + k n + k (n, k)= = . D1 n k ∗The author was partially supported by Universidad Sergio Arboleda under grant no. DII-262. the electronic journal of combinatorics 22(1) (2015), #P1.38 1 (1) Let 1 be the infinite lower triangular array defined by 1 = hn,k , where H H n,k∈N h i n > (1) 1(n k, k)= k , if n k; hn,k = D − (0, if n<k. Therefore, 1 is the Pascal matrix. This matrix has a lot of interesting properties, (see for instanceH [2, 5, 6, 7, 12, 23, 24]), one of them is the sum of the elements on the rising 1 diagonal is the Fibonacci sequence Fn, (sequence A000045) . If we add a third step, D1 = (1, 1), we obtain Delannoy paths; among other references see [1, 9, 10, 13, 22]. We denote by M2(n, k) the set of lattice paths from the point (0, 0) to the point (k,n), with step set S2 = H = (1, 0),V = (0, 1),D1 = (1, 1) . If (n, k)= M (n, k) , then it is known that { } D2 | 2 | k n n + k i (n, k)= − . D2 i n i=0 X The sequence 2(n,n) is called central Delannoy numbers (sequence A001850). Let D (2) 2 be the infinite lower triangular array defined by 2 = hn,k , where H H n,k∈N h i (2) 2(n k, k), if n > k; hn,k = D − (0, if n<k. This array is called Delannoy or tribonacci matrix. It satisfies that the sum of the elements on the rising diagonal is the tribonacci sequence tn (sequence A000073). ∗ A generalized Delannoy number 2(n, k) is the number of weighted lattice paths such that the steps H,V and D are labelledD 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]): k k n + k i ∗(n, k)= − ak−ibn−ici. (1) D2 i k i=0 X Cheon et al. [9] defined the generalized Delannoy triangle 2(a,b,c)=[dn,k]n,k∈N, where H ∗ 2(n k, k), if n > k; dn,k = D − (0, if n < k; 1Many integer sequences and their properties are expounded on The On-Line Encyclopedia of Integer Sequences [20]. the electronic journal of combinatorics 22(1) (2015), #P1.38 2 and, they studied this array by means of Riordan arrays. The first few terms of this triangle are 10 0 00 b a 0 00 b2 c +2ab a2 0 0 2(a,b,c)= b3 b(2c +3ab) a(2c +3ab) a3 0 . H b4 b2(3c +4ab) c2 +6abc +6a2b2 a2(3c +4ab) a4 . ··· . 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 (k) that the sum of the elements on its rising diagonal is the k-bonacci numbers n ? The k-bonacci numbers are defined by the recurrence F (k) = (k) + (k) + + (k) , n > 1, Fn Fn−1 Fn−2 ··· Fn−k with initial values (k) = (k) = = (k) = 0 and (k) = 1. In particular, we obtain F−1 F−2 ··· F−(k−1) F0 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 Sm = H = (1, 0),V = (0, 1),D = (1, 1),...,Dm = (1,m 1) , { 1 −1 − } where each step is labelled with weights a1,a2,...,am+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. 2 Riordan arrays A Riordan matrix L = [ln,k]n,k∈N is defined by a pair of generating functions g(z) = 1+ g z + g z2 + and f(z)= f z + f z2 + , where f = 0, such that the k-th column 1 2 ··· 1 2 ··· 1 6 satisfies n k ln,kz = g(z)(f(z)) . n>0 X the electronic journal of combinatorics 22(1) (2015), #P1.38 3 The first column being indexed by 0. From the definition, follows n k ln,k =[z ] g(z)(f(z)) , where [zn] is the coefficient operator. The matrix corresponding to the pair f(z),g(z) is denoted by (g(z),f(z)) or (g(z),f(z)). The product of two Riordan arrays (g(z),f(z)) and (h(z),l(Rz)) is defined by: (g(z),f(z)) (h(z),l(z))=(g(z)h (f(z)) ,l (f(z))) . ∗ 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),f(z))is(g(z),f(z))−∗1 = 1/ g f (z), f(z) , ◦ where f(z) is the compositional inverse of f(z). Example 1. The Pascal matrix 1 and the Delannoy matrix 2 are given by the following Riordan arrays, respectively. H H 1 z 1 z(1 + z) = , , = , . H1 1 z 1 z H2 1 z 1 z − − − − Example 2. ([9]) The generalized Delannoy array 2(a,b,c) has a Riordan array expres- sion given by H 1 a + cz (a,b,c)= ,z . H2 1 bz 1 bz − − The following theorem is known as the fundamental theorem of Riordan arrays or summation property (cf. [21]). Theorem 3. If [ln,k]n,k∈N = (g(z),f(z)) is a Riordan matrix. Then for any sequence hk k N { } ∈ n n ln,khk =[z ] g(z)h(f(z)), k X=0 where h(z) is the generating function of the sequence hk k N. { } ∈ > Let [ln,k]n,k∈N =(g(z),f(z)) be a Riordan array, then the elements ln−sk : k 0 are called the d-diagonal of (g(z),f(z)) (cf. [21]). { } 3 Generalized Tribonacci and Tribonacci-Lucas polynomial se- quence 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 trian- gular array. Then, we study its properties and we obtain the generalized tribonacci and the electronic journal of combinatorics 22(1) (2015), #P1.38 4 tribonacci-Lucas polynomial sequence. Let M3(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 S3 = H = (1, 0),V = (0, 1),D1 = (1, 1),D2 = (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 a2bcd. b a a d c Figure 1: Example of (a,b,c,d)-weighted path.