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

%I A036658 #23 Nov 27 2024 12:07:29
%S A036658 0,0,0,0,1,1,3,6,14,29,68,147,337,757,1734,3953,9113,20988,48645,
%T A036658 112909,263084,614201,1438001,3373253,7930660,18679005,44075988,
%U A036658 104173194,246604137,584620470,1387879434,3299067379,7851736348,18708682855,44627133541,106563177864
%N A036658 Number of n-node rooted unlabeled trees with exactly 3 edges at root and otherwise out-degree <= 2.
%H A036658 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A036658 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).
%F A036658 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)).
%p A036658 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;
%p A036658 N := 40: G036658 := series(x^3*CI3(G036656),x,N); A036658 := n->coeff(G036658,x,n);
%t A036658 terms = 35;
%t A036658 CI3[f_] := (1/3!)*(f^3 + 3*(f /. x -> x^2)*f + 2*(f /. x -> x^3));
%t A036658 G036656[_] = 0; Do[G036656[x_] = x + (1/2)*(G036656[x]^2 + G036656[x^2]) + O[x]^terms // Normal, terms];
%t A036658 G036658[x_] = x^3*CI3[G036656[x] - x] + O[x]^(terms+5);
%t A036658 Drop[CoefficientList[G036658[x], x], 5] (* _Jean-François Alcover_, Jan 24 2018, adapted from Maple *)
%Y A036658 Cf. A088326, A036656.
%K A036658 nonn,nice,easy
%O A036658 0,7
%A A036658 _N. J. A. Sloane_
%E A036658 Corrected by _N. J. A. Sloane_, May 03 2000