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Search: id:A000671
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| A000671 |
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Boron trees with n nodes = n-node rooted trees with deg <=3 at root and out-degree <=2 elsewhere.
(Formerly M1083 N0411)
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2
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| 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; listen)
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OFFSET
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0,4
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REFERENCES
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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).
S. J. Cyvin et al., Enumeration of constitutional isomers of polyenes, J. Molec. Structure (Theochem), 357 (1995), 255-261.
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).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..200
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
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FORMULA
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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.
A000671(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)).
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MAPLE
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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);
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CROSSREFS
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Sequence in context: A119342 A119268 A002989 this_sequence A157133 A120262 A013326
Adjacent sequences: A000668 A000669 A000670 this_sequence A000672 A000673 A000674
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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