Sums of powers of roots via Bell polynomials

Consider a function with a power series expansion and roots {x(n), 1 <= n < infinity}. We give two methods for obtaining the sums of powers of these roots, S-k = Sigma(infinity)(n=1)x(n)(k). The first is by expressing Girard's formula in terms of Bell polynomials. Explicit expressions are given for S-k for k <= 10. The second method is through generating functions.