Flajolet, Philippe; Odlyzko, Andrew Singularity analysis of generating functions. (English) Zbl 0712.05004 SIAM J. Discrete Math. 3, No. 2, 216-240 (1990). From the authors’ abstract: “This work presents a class of methods by which one can translate, on a term-by-term basis, an asymptotic expansion of a function around a dominant singularity into a corresponding asymptotic expansion for the Taylor coefficients of the function. This approach is based on contour integration using Cauchy’s formula and Hankel-like contours. It constitutes an alternative to either Darboux’s method or Tauberian theorems that appears to be well suited to combinatorial enumerations, and a few applications in this area are outlined.” Reviewer: R.C.Read Cited in 9 ReviewsCited in 286 Documents MathOverflow Questions: Recovering information for \(\sum_{n \leq x}a(n)\) from \(\sum_{n \geq 1}a(n)e^{-nx}\) MSC: 05A15 Exact enumeration problems, generating functions 40E05 Tauberian theorems Keywords:asymptotic analysis; generating functions; combinatorial enumeration; Tauberian theorems PDF BibTeX XML Cite \textit{P. Flajolet} and \textit{A. Odlyzko}, SIAM J. Discrete Math. 3, No. 2, 216--240 (1990; Zbl 0712.05004) Full Text: DOI Link OpenURL Digital Library of Mathematical Functions: Example ‣ §2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations Online Encyclopedia of Integer Sequences: B-trees of order 3 with n leaves. Sum over all n! permutations of n letters of maximum cycle length. Let y=(1-sqrt(1-4*z))/(1+sqrt(1-4*z)) denote the g.f. for the Catalan numbers (A000108); sequence has g.f. sum(k>=1, y^(2^k)/(1+y^(2^(k+1))) ).