1 research outputs found
A refinement of the formula for -ary trees and the Gould-Vandermonde's convolution
In this paper, we present an involution on some kind of colored -ary trees
which provides a combinatorial proof of a combinatorial sum involving the
generalized Catalan numbers . From the combinatorial sum, we refine the formula for
-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.Comment: Another informative title may be "A linear recurrence for generalized
Catalan numbers and its applications