  We investigate the asymptotic properties of higher-order binding corrections
to the one-loop self-energy of excited states in atomic hydrogen. We evaluate
the historically problematic A60 coefficient for all P states with principal
quantum numbers n <= 7 and D states with n <= 8 and find that a satisfactory
representation of the n-dependence of the coefficients requires a
three-parameter fit. For the high-energy contribution to A60, we find exact
formulas. The results obtained are relevant for the interpretation of
high-precision laser spectrocopic measurements.
