We derive an asymptotic expansion as n→∞ for a large range of coefficients of (f(z))n, where f(z) is a power series satisfying |f(z)|<f(|z|) for z∈ℂ,
z∉ℝ+. When f is a polynomial and the two
smallest and the two largest exponents appearing in f are
consecutive integers, we use the expansion to generalize results
of Odlyzko and Richmond (1985) on log concavity of polynomials,
and we prove that a power of f has only positive coefficients.