$k$ -叉树公式的改进和Gould-Vandermonde卷积
A refinement of the formula for $k$-ary trees and the Gould-Vandermonde's convolution
摘要：
本文给出了一类有色$k$ -叉树上的对合，从而给出了广义Catalan数$C_{k,γ}(n)=\fracγ{k n+γ}{k n+γ\choose n}$的组合和的组合证明.从组合和出发，我们改进了$k$ -叉树的公式，得到了广义Catalan数的生成函数的一个隐式公式，它显然蕴含了Gould推广的Vandermonde型卷积.此外，我们还得到了一个组合和涉及的向量推广的Catalan数的扩展我们的对合。
英文摘要：
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 $C_{k,γ}(n)=\fracγ{k n+γ}{k n+γ\choose n}$. 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.
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