A refinement of the formula for $k$-ary trees and the Gould-Vandermonde's convolution

TL;DR

This paper introduces a combinatorial involution on colored k-ary trees to refine formulas for generalized Catalan numbers, providing new proofs, generating functions, and a Gould-Vandermonde convolution extension.

Contribution

It offers a novel combinatorial involution to prove a sum involving generalized Catalan numbers and refines the formula for k-ary trees with a new generating function.

Findings

Provides a combinatorial proof of a sum involving generalized Catalan numbers

Refines the formula for k-ary trees

Derives an implicit generating function and extends Gould-Vandermonde convolution

Abstract

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 $C_{k,\gamma}(n)=\frac{\gamma}{k n+\gamma}{k n+\gamma\choose n}$. 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.