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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
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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. 454).
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 527.
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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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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MATHEMATICA
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m = 45; (* T = G000598 *) T[_] = 0; Do[T[z_] = 1 + z*(T[z]^3/6 + T[z^2]*T[z]/2 + T[z^3]/3) + O[z]^m // Normal, m];
G000678[z_] = z*(T[z]^4/24 + T[z^2]*T[z]^2/4 + T[z^2]^2/8 + T[z]*T[z^3]/3 + T[z^4]/4) + O[z]^m;
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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