We study the renormalized perturbation theory of the single-impurity Anderson model, particularly the high-order terms in the expansion of the self-energy in powers of the renormalized coupling
˜
U
. Though the presence of counterterms in the renormalized theory may appear to complicate the diagrammatics, we show how these can be seamlessly accommodated by carrying out the calculation order-by-order in terms of skeleton diagrams. We describe how the diagrams pertinent to the renormalized self-energy and four vertex can be automatically generated, translated into integrals, and numerically integrated. To maximize the efficiency of our approach we introduce a generalized
k
-particle/hole propagator, which is used to analytically simplify the resultant integrals and reduce the dimensionality of the integration. We present results for the self-energy and spectral density to fifth order in
˜
U
, for various values of the model asymmetry, and compare them to a numerical renormalization group calculation.