This note is concerned with the computation of power-series-expansion coefficients of a given function. It is demonstrated that the mathematical formulation of Abate and Dubner (1968) and the improved FFT-based Laplace inversion algorithm of Hwang et al. (1991) can be combined to give an efficient algorithm for determining accurate coefficients of a power series expansion. The algorithm is particularly suitable for being programmed on a modern parallel computing environment. As an application, the algorithm is employed to find approximate continuous inverse functions of Laplace transforms.