A refinement of the formula for \(k\)-ary trees and the Gould-Vandermonde's convolution
The electronic journal of combinatorics, Tome 15 (2008)
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
Mots-clés : combinatorial sum, generalized Catalan numbers, k-ary trees, generating function, Vandermonde type convolution, vector generalization of Catalan numbers
@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/}
}
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
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