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%I M1171 N0448 #37 Feb 09 2018 21:45:32
%S 0,1,1,2,4,9,18,42,96,229,549,1347,3326,8330,21000,53407,136639,
%T 351757,909962,2365146,6172068,16166991,42488077,112004630,296080425,
%U 784688263,2084521232,5549613097,14804572332,39568107511,105938822149
%N Number of carbon (rooted) trees with n carbon atoms = unordered 4-tuples of ternary trees.
%D 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 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 527.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H N. J. A. Sloane, <a href="/A000678/b000678.txt">Table of n, a(n) for n = 0..60</a>
%H Jean-François Alcover, <a href="/A000602/a000602_1.txt">Mathematica program translated from N. J. A. Sloane's Maple program for A000022, A000200, A000598, A000602, A000678</a>
%H G. Polya, <a href="http://dx.doi.org/10.1524/zkri.1936.93.1.415">Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen</a>, Zeit. f. Kristall., 93 (1936), 415-443; line 10 of Table I.
%H G. Polya, <a href="/A000598/a000598_3.pdf">Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen</a>, Zeit. f. Kristall., 93 (1936), 415-443; Table I, line 10 (Annotated scanned copy)
%H E. M. Rains and N. J. A. Sloane, <a href="https://cs.uwaterloo.ca/journals/JIS/cayley.html">On Cayley's Enumeration of Alkanes (or 4-Valent Trees)</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
%H R. C. Read, <a href="/A000598/a000598.pdf">The Enumeration of Acyclic Chemical Compounds</a>, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See g.f. called P(x) on p. 28, 37.
%H N. J. A. Sloane, <a href="/A000602/a000602.txt">Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F G.f.: A(x) = x*cycle_index(S4, B(x)), B(x) = g.f. for A000598.
%e z+z^2+2*z^3+4*z^4+9*z^5+18*z^6+42*z^7+...
%p Let T_i(z) = g.f. for ternary trees of height at most i.
%p 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 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 (this Maple program continues in A000022.)
%t 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];
%t 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;
%t CoefficientList[G000678[z], z] (* _Jean-François Alcover_, Jan 11 2018, after _N. J. A. Sloane_ *)
%K nonn,easy,nice
%O 0,4
%A _N. J. A. Sloane_, E. M. Rains (rains(AT)caltech.edu)
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