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MATHEMATICA
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n = 30; (* algorithm from Rains and Sloane *)
m = 7; (* maximum degree of node *)
CIm[f_, h_, x_] = SymmetricGroupIndex[m-1, x] /. x[i_] -> f[h, x^i];
CI[f_, h_, x_] = SymmetricGroupIndex[m, x] /. x[i_] -> f[h, x^i];
T[-1, z_] := 1; T[h_, z_] := T[h, z] = Table[z^k, {k, 0, n}] .
Take[CoefficientList[z^(n+1) + 1 + CIm[T, h-1, z] z, z], n+1];
ReplacePart[Sum[Take[CoefficientList[z^(n+1) + CI[T, h-1, z] z - CI[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}],
1->1] (* end of original program *)
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 = 6; (* m = maximum children *) n = 40;
gf[x_] = 1 + Sum[b[j-1, j-1, m, m]x^j, {j, 1, n}]; (* G.f. for A036722 *)
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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