A generalized version of Groeneveld's convergence criterion for the virial expansion and generating functionals for weighted $2$-connected graphs is proven. The criterion works for inhomogeneous systems and yields bounds for the density expansions of the correlation functions $ρ_s$ (a.k.a. distribution functions or factorial moment measures) of grand-canonical Gibbs measures with pairwise interactions. The proof is based on recurrence relations for graph weights related to the Kirkwood-Salsburg integral equation for correlation functions. The proof does not use an inversion of the density-activity expansion, however a Moebius inversion on the lattice of set partitions enters the derivation of the recurrence relations.