A036658 Number of n-node rooted unlabeled trees with exactly 3 edges at root and otherwise out-degree <= 2.

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, 106563177864

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Links

• Index entries for sequences related to rooted trees

Crossrefs

Cf. A088326, A036656.

Programs

Maple

CI2 := proc(f) (1/2)*(f^2+subs(x=x^2,f)); end; CI3 := proc(f) (1/6)*(f^3+3*subs(x=x^2,f)*f+2*subs(x=x^3,f)); end;
N := 40: G036658 := series(x^3*CI3(G036656),x,N); A036658 := n->coeff(G036658,x,n);

Mathematica

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 *)

Formula

Let G036656(x) = g.f. for A036656. G.f.: x^3*cycle_index(S3, G036656), where cycle_index(Sk, f) means apply the cycle index for the symmetric group S_k to f(x).

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)).

Extensions

Corrected by N. J. A. Sloane, May 03 2000