Geodesic
A refinement of the formula for \(k\)-ary trees and the Gould-Vandermonde's convolution
The electronic journal of combinatorics, Tome 15 (2008)

Voir la notice de l'article provenant de la source The Electronic Journal of Combinatorics website

Zbl arXiv EuDML
In this paper, we present an involution on some kind of colored $k$-ary trees which provides a combinatorial proof of a combinatorial sum involving the generalized Catalan numbers $C_{k,\gamma}(n)={\gamma\over k n+\gamma}{k n+\gamma\choose n}$. From the combinatorial sum, we refine the formula for $k$-ary trees and obtain an implicit formula for the generating function of the generalized Catalan numbers which obviously implies a Vandermonde type convolution generalized by Gould. Furthermore, we also obtain a combinatorial sum involving a vector generalization of the Catalan numbers by an extension of our involution.
DOI : 10.37236/776
Classification : 05A19, 05A15, 05C05
Mots-clés : combinatorial sum, generalized Catalan numbers, k-ary trees, generating function, Vandermonde type convolution, vector generalization of Catalan numbers 
Ricky X. F. Chen. A refinement of the formula for \(k\)-ary trees and the Gould-Vandermonde's convolution. The electronic journal of combinatorics, Tome 15 (2008). doi: 10.37236/776
@article{10_37236_776,
     author = {Ricky X. F. Chen},
     title = {A refinement of the formula for \(k\)-ary trees and the {Gould-Vandermonde's} convolution},
     journal = {The electronic journal of combinatorics},
     year = {2008},
     volume = {15},
     doi = {10.37236/776},
     zbl = {1159.05004},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/776/}
}
TY  - JOUR
AU  - Ricky X. F. Chen
TI  - A refinement of the formula for \(k\)-ary trees and the Gould-Vandermonde's convolution
JO  - The electronic journal of combinatorics
PY  - 2008
VL  - 15
UR  - http://geodesic.mathdoc.fr/articles/10.37236/776/
DO  - 10.37236/776
ID  - 10_37236_776
ER  - 
%0 Journal Article
%A Ricky X. F. Chen
%T A refinement of the formula for \(k\)-ary trees and the Gould-Vandermonde's convolution
%J The electronic journal of combinatorics
%D 2008
%V 15
%U http://geodesic.mathdoc.fr/articles/10.37236/776/
%R 10.37236/776
%F 10_37236_776

Cité par Sources :