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We ask for $n\\in\\mathbb{N}$ such that the equation $G_n(x)=g\\circ h$ is satisfied for a polynomial $g\\in\\mathbb{C}[x]$ with deg$g=m$ and some polynomial $h\\in\\mathbb{C}[x]$ with deg$h>1$. We prove that for all but finitely many $n$ these decompositions can be described in \"finite terms\" coming from a generic decomposition parameterized by an algebraic variety. 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