In this article, we introduce a family of weighted lattice paths, whose step set is $\{H=(1,0), V=(0,1), D_1=(1,1), \dots, D_{m-1}=(1,m-1)\}$. 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 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.
@article{10_37236_4618,
author = {Jos\'e L. Ram{\'\i}rez and V{\'\i}ctor F. Sirvent},
title = {A generalization of the \(k\)-bonacci sequence from {Riordan} arrays},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {1},
doi = {10.37236/4618},
zbl = {1308.11016},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4618/}
}
TY - JOUR
AU - José L. Ramírez
AU - Víctor F. Sirvent
TI - A generalization of the \(k\)-bonacci sequence from Riordan arrays
JO - The electronic journal of combinatorics
PY - 2015
VL - 22
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/4618/
DO - 10.37236/4618
ID - 10_37236_4618
ER -
%0 Journal Article
%A José L. Ramírez
%A Víctor F. Sirvent
%T A generalization of the \(k\)-bonacci sequence from Riordan arrays
%J The electronic journal of combinatorics
%D 2015
%V 22
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/4618/
%R 10.37236/4618
%F 10_37236_4618
José L. Ramírez; Víctor F. Sirvent. A generalization of the \(k\)-bonacci sequence from Riordan arrays. The electronic journal of combinatorics, Tome 22 (2015) no. 1. doi: 10.37236/4618
