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Number of n-node rooted unlabeled trees with exactly 3 edges at root and otherwise out-degree <= <= 2.

E.g. ., cycle_index(S2, f) = (1/2!)*(f^2+subs(x=x^2, f), cycle_index(S3, f) = (1/3!)*(f^3+3*subs(x=x^2, f)*f+2*subs(x=x^3, f)).

_N. J. A. Sloane_._

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Corrected May 03 2000 - _by _N. J. A. Sloane_._, May 03 2000

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terms = 35;

CI3[f_] := (1/3!)*(f^3 + 3*(f /. x -> x^2)*f + 2*(f /. x -> x^3));

G036656[_] = 0; Do[G036656[x_] = x + (1/2)*(G036656[x]^2 + G036656[x^2]) + O[x]^terms // Normal, terms];

G036658[x_] = x^3*CI3[G036656[x] - x] + O[x]^(terms+5);

Drop[CoefficientList[G036658[x], x], 5] (* Jean-François Alcover, Jan 24 2018, adapted from Maple *)

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0, 0, 0, 0, 1, 1, 3, 6, 14, 29, 68, 147, 337, 757, 1734, 3953, 9113, 20988, 48645, 112909, 263084, 614201, 1438001, 3373253, 7930660, 18679005, 44075988, 104173194, 246604137, 584620470, 1387879434, 3299067379, 7851736348, 18708682855, 44627133541

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Alois P. Heinz: ...