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A refinement of the formula for $k$-ary trees and the Gould-Vandermondes convolution

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 نشر من قبل Ricky Xiaofeng Chen
 تاريخ النشر 2019
  مجال البحث
والبحث باللغة English
 تأليف Ricky X. F. Chen




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