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A000678 Number of carbon (rooted) trees with n carbon atoms = unordered 4-tuples of ternary trees.
(Formerly M1171 N0448)
5
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; refs; listen; history; text; internal format)
OFFSET

0,4

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. 454).

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 527.

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.

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

N. J. A. Sloane, Table of n, a(n) for n = 0..60

Jean-Fran├žois Alcover, Mathematica program translated from N. J. A. Sloane's Maple program for A000022, A000200, A000598, A000602, A000678

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

FORMULA

G.f.: A(x) = x*cycle_index(S4, B(x)), B(x) = g.f. for A000598.

EXAMPLE

z+z^2+2*z^3+4*z^4+9*z^5+18*z^6+42*z^7+...

MAPLE

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.)

CROSSREFS

Sequence in context: A094291 A026765 A032175 * A081490 A129784 A125050

Adjacent sequences:  A000675 A000676 A000677 * A000679 A000680 A000681

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, E. M. Rains (rains(AT)caltech.edu)

STATUS

approved

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Last modified September 17 19:32 EDT 2014. Contains 246868 sequences.