We find a formula for the asymptotics of the coefficients of a generating function of the form, H(z1,z1,...,zd)−β, as the indices approach infinity in a fixed ratio. Then, we look at how this formula can be applied to generating functions that enumerate the possible structures into which RNA sequences can fold. This work relies on the techniques in multivariate analytic combinatorics developed by Pemantle and Wilson. We combine the multivariate Cauchy integral formula with explicit contour deformations to compute the asymptotic formula. A challenge of using the formula is correctly identifying the points which contribute to asymptotics.