An asymptotic formula for $\exp(\frac{x}{1-x})$
Commun. Korean Math. Soc. 2002 Vol. 17, No. 2, 363-370
Printed June 1, 2002
Junho Song, Changwoo Lee
University of Seoul, University of Seoul
Abstract : We show that $G(x)=e^{x/(1-x)}-1$ is the exponential generating function for the labeled digraphs whose weak components are transitive tournaments and derive both a recursive formula and an explicit formula for the number of them on $n$ vertices. Moreover, we investigate the asymptotic behavior for the coefficients of $G(x)$ using Hayman's method.
Keywords : transitive tournament, exponential generating function, recursive formula, asymptotics, admissible, Hayman's method
MSC numbers : 05C30, 05C20
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