%I A000678 M1171 N0448
%S A000678 0,1,1,2,4,9,18,42,96,229,549,1347,3326,8330,21000,53407,136639,
%T A000678 351757,909962,2365146,6172068,16166991,42488077,112004630,296080425,
%U A000678 784688263,2084521232,5549613097,14804572332,39568107511,105938822149
%N A000678 Number of carbon (rooted) trees with n carbon atoms = unordered 4-tuples 
               of ternary trees.
%D A000678 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).
%D A000678 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 
               2004; p. 527.
%D A000678 G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer 
               Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; line 10 of 
               Table I.
%D A000678 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.
%D A000678 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000678 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000678 N. J. A. Sloane, <a href="b000678.txt">Table of n, a(n) for n = 0..60</
               a>
%H A000678 E. M. Rains and N. J. A. Sloane, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">On Cayley's Enumeration of Alkanes (or 4-Valent 
               Trees)</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
%H A000678 N. J. A. Sloane, <a href="a000602.txt">Maple program and first 60 terms 
               for A000022, A000200, A000598, A000602, A000678</a>
%H A000678 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%H A000678 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%F A000678 G.f.: A(x) = x*cycle_index(S4, B(x)), B(x) = g.f. for A000598.
%e A000678 z+z^2+2*z^3+4*z^4+9*z^5+18*z^6+42*z^7+...
%p A000678 Let T_i(z) = g.f. for ternary trees of height at most i.
%p A000678 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
%p A000678 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.
%p A000678 (this Maple program continues in A000022.)
%Y A000678 Sequence in context: A094291 A026765 A032175 this_sequence A081490 A129784 
               A125050
%Y A000678 Adjacent sequences: A000675 A000676 A000677 this_sequence A000679 A000680 
               A000681
%K A000678 nonn,easy,nice
%O A000678 0,4
%A A000678 N. J. A. Sloane (njas(AT)research.att.com), E. M. Rains (rains(AT)caltech.edu)