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MATHEMATICA
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n = 20; (* algorithm from Rains and Sloane *)
S5[f_, h_, x_] := f[h, x]^5/120 + f[h, x]^3 f[h, x^2]/12 + f[h, x]^2 f[h, x^3]/6 + f[h, x] f[h, x^2]^2/8 + f[h, x] f[h, x^4]/4 + f[h, x^2] f[h, x^3]/6 + f[h, x^5]/5;
S6[f_, h_, x_] := f[h, x]^6/720 + f[h, x]^4 f[h, x^2]/48 + f[h, x]^3 f[h, x^3]/18 + f[h, x]^2 f[h, x^2]^2/16 + f[h, x]^2 f[h, x^4]/8 + f[h, x] f[h, x^2] f[h, x^3]/6 + f[h, x] f[h, x^5]/5 + f[h, x^2]^3/48 + f[h, x^2] f[h, x^4]/8 + f[h, x^3]^2/18 + f[h, x^6]/6;
T[-1, z_] := 1; T[h_, z_] := T[h, z] = Table[z^k, {k, 0, n}].Take[CoefficientList[z^(n+1) + 1 + S5[T, h-1, z]z, z], n+1];
Sum[Take[CoefficientList[z^(n+1) + S6[T, h-1, z]z - S6[T, h-2, z]z - (T[h-1, z] - T[h-2, z]) (T[h-1, z]-1), z], n+1], {h, 1, n/2}] + PadRight[{0, 1}, n+1] + Sum[Take[CoefficientList[z^(n+1) + (T[h, z] - T[h-1, z])^2/2 + (T[h, z^2] - T[h-1, z^2])/2, z], n+1], {h, 0, n/2}] (* Robert A. Russell, Sep 15 2018 *)
b[n_, i_, t_, k_] := b[n, i, t, k] = If[i<1, 0, Sum[Binomial[b[i-1, i-1,
k, k] + j-1, j]* b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]];
b[0, i_, t_, k_] = 1; m = 5; (* m = maximum children *) n = 40;
gf[x_] = 1 + Sum[b[j-1, j-1, m, m]x^j, {j, 1, n}]; (* G.f. for A036721 *)
ci[x_] = SymmetricGroupIndex[m+1, x] /. x[i_] -> gf[x^i];
CoefficientList[Normal[Series[gf[x] - (gf[x]^2 - gf[x^2])/2 + x ci[x],
{x, 0, n}]], x] (* Robert A. Russell, Jan 19 2023 *)
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