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| MATHEMATICA
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max = 30; sol[m_] := (c[0] = 0; f81[x_] := Sum[c[k]*x^k, {k, 0, m}]; cc = CoefficientList[Series[f81[x] - x*Exp[Sum[f81[x^k]/k, {k, 1, m}]], {x, 0, m}], x]; sc = First[Solve[Thread[cc == 0]]]; t[x_] := f81[x] /. sc; f[x_] := Sum[b[k]*x^k, {k, 0, m}]; b[0] = 0; bb = CoefficientList[Series[f[x] - t[x]*(1 + f[x] - f[x^2]), {x, 0, m}], x]; sb = First[Solve[Thread[bb == 0]]]; Apply[Set, sc, {1}]; Apply[Set, sb, {1}]); Do[sol[m], {m, 5, max, 5}]; Drop[CoefficientList[f[x] /. sb, x], 1] (* Jean-François Alcover, Sep 30 2011 *)
terms = 30; (* T = g.f. of A000081 *)
T[x_] = 0; Do[T[x_] = x*Exp[Sum[ T[x^k]/k, {k, 1, terms}]] + O[x]^(terms+1) // Normal, terms+1];
A[_] = 0; Do[A[x_] = T[x]*(1 + A[x] - A[x^2]) + O[x]^(terms+1) // Normal,
terms+1];
Drop[CoefficientList[A[x], x] , 1] (* Jean-François Alcover, Sep 30 2011, updated Jan 11 2018 *)
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