$$f(n) = \dfrac{(-a)^{n-1} \Gamma(n - 1/a)}{\Gamma(1 - 1/a)}$$
Use Stirling's series for the asymptotic approximation of $\ln(\Gamma(n - 1/a))$, or one of its variants that you can find at http://en.wikipedia.org/wiki/Stirling%27s_approximation