A Generalization of the k-Bonacci Sequence from Riordan Arrays

In this article, we introduce a family of weighted lattice paths, whose step set is {H = (1,0), V = (0,1), D-1 = (1,1),..., Dm-1 = (1, m - 1)}. 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 i 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.