  We find sufficient conditions for a discrete sequence to be interpolating or
sampling for certain generalized Bergman spaces on open Riemann surfaces. As in
previous work of Bendtsson, Ortega-Cerda, Seip, Wallsten and others, our
conditions for interpolation and sampling are as follows: If a certain upper
density of the sequence has value less that 1, then the sequence is
interpolating, while if a certain lower density has value greater than 1, then
the sequence is sampling.
  However, unlike previous work, we construct an infinite number of densities,
naturally parameterized by $L^1_{loc} ([0,\infty))$. These densities provide
(different) sufficient conditions for interpolation and sampling. As a
consequence of this flexibility, we obtain new results even in the classical
cases of the Bargmann-Fock space (of entire functions integrable with respect
to a Gaussian weight) and the classical Bergman space of L^2 holomorphic
functions on the unit disk.
