A refinement of the formula for \(k\)-ary trees and the Gould-Vandermonde's convolution (Q1010768)

Summary: 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.