In this paper we take inspiration from Weit'z algorithm for approximating the independence polynomial to provide a new algorithm for computing the coefficients of the Taylor series of the logarithm of the independence polynomial. Hereby we provide a clear connections between Barvinok's interpolation method and Weitz's algorithm. Our algorithm easily extends to other graph polynomials and partition functions and we illustrate this by applying it to the chromatic polynomial and to the graph homomorphism partition function. Our approach arguably yields a simpler and more transparent algorithm than the algorithm of Patel and the second author. As an application of our algorithmic approach we moreover derive, using the interpolation method, a deterministic $O(n(m/\varepsilon)^{7})$-time algorithm that on input of an $n$-vertex and $m$-edge graph of minimum degree at least $3$ and $\varepsilon>0$ approximately computes the number of sink-free orientations of $G$ up to a multiplicative $\exp(\varepsilon)$ factor.