Coefficients of functional compositions often grow smoothly
Summary: The coefficients of a power series \(A(x)\) are smooth if \(a_{n-1}/a_n\) approaches a limit. If \(A(x)=F(G(x))\) and \(f_n^{1/n}\) approaches a limit, then the coefficients of \(A(x)\) are often smooth. We use this to show that the coefficients of the exponential generating function for graphs embeddable on a given surface are smooth, settling a problem of \textit{C.McDiarmid}, \textit{A. Steger}, and \textit{D.J.A. Welsh} [''Random planar graphs,'' J. Comb. Theory, Ser. B. 93, No.\,2, 187--205 (2005; Zbl 1056.05128)].
scientific article; zbMATH DE number 5540926