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Search: id:A000678
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| A000678 |
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Number of carbon (rooted) trees with n carbon atoms = unordered 4-tuples of ternary trees.
(Formerly M1171 N0448)
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5
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| 0, 1, 1, 2, 4, 9, 18, 42, 96, 229, 549, 1347, 3326, 8330, 21000, 53407, 136639, 351757, 909962, 2365146, 6172068, 16166991, 42488077, 112004630, 296080425, 784688263, 2084521232, 5549613097, 14804572332, 39568107511, 105938822149
(list; graph; listen)
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OFFSET
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0,4
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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. 454).
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; line 10 of Table I.
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see g.f. called P(x) on p. 28, 37.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 527.
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 0..60
E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
N. J. A. Sloane, Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678
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*cycle_index(S4, B(x)), B(x) = g.f. for A000598.
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EXAMPLE
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z+z^2+2*z^3+4*z^4+9*z^5+18*z^6+42*z^7+...
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MAPLE
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Let T_i(z) = g.f. for ternary trees of height at most i.
N := 45; G000598 := 0: i := 0: while i<(N+1) do G000598 := series(1+z*(G000598^3/6+subs(z=z^2, G000598)*G000598/2+subs(z=z^3, G000598)/3)+O(z^(N+1)), z, N+1): t[ i ] := G000598: i := i+1: od: # G000598 = g.f. for A000598
i := 0: while i<N+1 do T := t[ i ]: G000678 := series(z*(T^4/24+subs(z=z^2, T)*T^2/4+subs(z=z^2, T)^2/8+T*subs(z=z^3, T)/3+subs(z=z^4, T)/4)+O(z^(N+1)), z, N+1): q[ i ] := G000678: i := i+1: od: A000678 := n->coeff(G000678, z, n); # G000678 = g.f. for A000678.
(this Maple program continues in A000022.)
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CROSSREFS
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Sequence in context: A094291 A026765 A032175 this_sequence A081490 A129784 A125050
Adjacent sequences: A000675 A000676 A000677 this_sequence A000679 A000680 A000681
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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), E. M. Rains (rains(AT)caltech.edu)
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