s[m_, p_, n_] :=
  CycleIndexPolynomial[SymmetricGroup[m],
   Table[ComposeSeries[p, x^i + O[x]^(n + 1)], {i, m}]];
G000598[n_] := Nest[1 + x s[3, #, n] &, 1 + O[x], n];
G000602[n_] :=
  x s[4, #, n] - s[2, # - 1, n] +
     ComposeSeries[#, x^2 + O[x]^(n + 1)] &[G000598[n]];
A000602[n_] := SeriesCoefficient[G000602[n], n];
A000602List[n_] := CoefficientList[G000602[n], x];
Grid@Transpose@{Range[0, 200], A000602List@200}