%I A000671 M1083 N0411
%S A000671 0,1,1,2,4,7,14,29,60,127,275,598,1320,2936,6584,14858,33744,76999,
%T A000671 176557,406456,939241,2177573,5064150,11809632,27610937,64705623,
%U A000671 151966597,357623905,843176524,1991439229,4711115672,11162025770
%N A000671 Boron trees with n nodes = n-node rooted trees with deg <=3 at root and 
               out-degree <=2 elsewhere.
%D A000671 A. Cayley, On the analytical forms called trees, with application to 
               the theory of chemical combinations, Reports British Assoc. Advance. 
               Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 450).
%D A000671 S. J. Cyvin et al., Enumeration of constitutional isomers of polyenes, 
               J. Molec. Structure (Theochem), 357 (1995), 255-261.
%D A000671 R. C. Read, personal communication.
%D A000671 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000671 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000671 T. D. Noe, <a href="b000671.txt">Table of n, a(n) for n=0..200</a>
%H A000671 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%H A000671 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%F A000671 G.f.: A(x) = x*(1/3!)*(f^3+3*subs(x=x^2, f)*f+2*subs(x=x^3, f)), where 
               f = G001190(x)/x, G001190 = g.f. for A001190.
%F A000671 A000671(n) = A001190(n) + A036657(n) + A036658(n).
%F A000671 Another g.f.: let B0(x) = 1+x, G036656(x) = g.f. for A036656, G036657(x) 
               = g.f. for A036657.
%F A000671 Then g.f. = x*(cycle_index(S3, B0)+cycle_index(S3, G036656)+cycle_index(S3, 
               G036657)+cycle_index(S2, B0)*(G036656+G036657)+cycle_index(S2, G036656)*(G036657+B0)+cycle_index(S2, 
               G036657)*(B0+G036656)+B0*G036656*G036657), where cycle_index(Sk, 
               f) means apply the cycle index for the symmetric group S_k to f(x).
%F A000671 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 A000671 N := 40: t1 := G001190/x: G000671 := series(x*(1/3!)*(t1^3+3*subs(x=x^2,
               t1)*t1+2*subs(x=x^3,t1)), x, N); A000671 := n->coeff(G000671,x,n);
%p A000671 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 A000671 N := 40: B0 := series(1 + x,x,N): G000671 := series(x*(CI3(B0) + CI3(G036656) 
               + CI3(G036657) + CI2(B0)*(G036656 + G036657) + CI2(G036656)*(G036657 
               + B0) + CI2(G036657)*(B0 + G036656) + B0*G036656*G036657),x,N); A036658 
               := n->coeff(G036658,x,n);
%Y A000671 Sequence in context: A119342 A119268 A002989 this_sequence A157133 A120262 
               A013326
%Y A000671 Adjacent sequences: A000668 A000669 A000670 this_sequence A000672 A000673 
               A000674
%K A000671 nonn,easy,nice
%O A000671 0,4
%A A000671 N. J. A. Sloane (njas(AT)research.att.com).