On asymptotics of solution of nonlinear difference equation of convolution type

Nonlinear difference equations appear in many problems of probability theory, computer science, and combinatorics. In this paper, a nonlinear difference equation of the convolution type with parameters is considered. Asymptotics of solutions of such equations are used for the enumeration of labeled connected graphs. To obtain the asymptotics, we apply Bender's theorem for the coefficients of formal power series.