  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 $\tilde{U}$. Though the presence of
counter-terms 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
$\tilde{U}$, for various values of the model asymmetry, and compare them to a
Numerical Renormalization Group calculation.
