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 A000671 Boron trees with n nodes = n-node rooted trees with deg <=3 at root and out-degree <=2 elsewhere. (Formerly M1083 N0411) 4
 0, 1, 1, 2, 4, 7, 14, 29, 60, 127, 275, 598, 1320, 2936, 6584, 14858, 33744, 76999, 176557, 406456, 939241, 2177573, 5064150, 11809632, 27610937, 64705623, 151966597, 357623905, 843176524, 1991439229, 4711115672, 11162025770 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The subsequence of primes begins: 2, 7, 29, 127, 176557, 2177573, 151966597. REFERENCES 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). R. C. Read, personal communication. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n=0..200 S. J. Cyvin, J. Brunvoll, B. N. Cyvin, Enumeration of constitutional isomers of polyenes, J. Molec. Struct. (Theochem) 357, no. 3 (1995) 255-261. R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976 FORMULA 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. a(n) = A001190(n) + A036657(n) + A036658(n). Another g.f.: let B0(x) = 1+x, G036656(x) = g.f. for A036656, G036657(x) = g.f. for A036657. 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). 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)). MAPLE 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); 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: 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); MATHEMATICA 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]; f[x_] = B[x]/x; A[x_] = x*(1/3!)*(f[x]^3 + 3*f[x^2]*f[x] + 2*f[x^3]) + O[x]^terms; CoefficientList[A[x], x] (* Jean-François Alcover, May 29 2012, from first g.f., updated Jan 10 2018 *) CROSSREFS Sequence in context: A002989 A293336 A321401 * A199888 A157133 A202850 Adjacent sequences:  A000668 A000669 A000670 * A000672 A000673 A000674 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified December 11 21:28 EST 2018. Contains 318052 sequences. (Running on oeis4.)