genus[Q_, n_Integer] :=
  Module[{z, x},
   SymmetricReduction[
      SeriesCoefficient[
       Product[ComposeSeries[Series[Q[z], {z, 0, n}],
         Series[x[i] z, {z, 0, n}]], {i, 1, n}], n],
      Table[x[i], {i, 1, n}], Table[Subscript[c, i], {i, 1, n}]][[
     1]] // FactorTerms];
AgenusTotal[n_Integer] :=
  Total[Table[
    genus[(Sqrt[#]/2)/Sinh[Sqrt[#]/2] &, i] /. c -> p, {i, 0, n}]];

Series[Series[
  Series[Sqrt[AgenusTotal[3]], {Subscript[p, 1], 0, 5}], {Subscript[p,
     2], 0, 5}], {Subscript[p, 3], 0, 5}]
SeriesCoefficient[
  SeriesCoefficient[
   SeriesCoefficient[
    Series[Series[
      Series[Sqrt[AgenusTotal[3]], {Subscript[p, 1], 0,
        5}], {Subscript[p, 2], 0, 5}], {Subscript[p, 3], 0, 5}], 2],
   2], 1];