%I M1083 N0411 #54 May 23 2023 21:00:24
%S 0,1,1,2,4,7,14,29,60,127,275,598,1320,2936,6584,14858,33744,76999,
%T 176557,406456,939241,2177573,5064150,11809632,27610937,64705623,
%U 151966597,357623905,843176524,1991439229,4711115672,11162025770,26484061667,62923251955
%N Number of boron trees with n nodes, i.e. n-node rooted trees with degree <= 3 at root and out-degree <= 2 elsewhere.
%C The subsequence of primes begins: 2, 7, 29, 127, 176557, 2177573, 151966597.
%D 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 R. C. Read, personal communication.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000671/b000671.txt">Table of n, a(n) for n = 0..200</a>
%H S. J. Cyvin, J. Brunvoll, and B. N. Cyvin, <a href="http://dx.doi.org/10.1016/0166-1280(95)04329-6"> Enumeration of constitutional isomers of polyenes</a>, J. Molec. Struct. (Theochem) 357, no. 3 (1995) 255-261.
%H R. C. Read, <a href="/A000684/a000684_1.pdf">Letter to N. J. A. Sloane, Oct. 29, 1976</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F 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 a(n) = A001190(n) + A036657(n) + A036658(n).
%F Another g.f.: let B0(x) = 1+x, G036656(x) = g.f. for A036656, G036657(x) = g.f. for A036657.
%F 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 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 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 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 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);
%t terms = 32; (* B = g.f. for A001190 *) B[_] = 0; Do[B[x_] = x + (1/2)*(B[x]^2 + B[x^2]) + O[x]^terms // Normal, terms];
%t f[x_] = B[x]/x;
%t A[x_] = x*(1/3!)*(f[x]^3 + 3*f[x^2]*f[x] + 2*f[x^3]) + O[x]^terms;
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, May 29 2012, from first g.f., updated Jan 10 2018 *)
%Y Cf. A001190, A036657, A036658.
%K nonn,easy,nice
%O 0,4
%A _N. J. A. Sloane_