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%I M2490 N0987 #78 Dec 06 2020 14:02:13
%S 1,1,3,5,13,27,66,153,377,914,2281,5690,14397,36564,93650,240916,
%T 623338,1619346,4224993,11062046,29062341,76581151,202365823,
%U 536113477,1423665699,3788843391,10103901486,26995498151,72253682560,193706542776
%N Number of ethylene derivatives with n carbon atoms.
%C Number of structural isomers of alkenes C_n H_{2n} with n carbon atoms.
%C Number of unicyclic graphs of n nodes where a double-edge replaces the cycle, [A217781], end-points of the double-edge of out-degrees <= 2, other nodes having out-degrees <= 3.
%C Number of rooted trees on n+1 nodes where the root has degree 2, the 2 children of the root have out-degrees <= 2, and the other nodes have out-degrees <= 3.
%C See illustration of initial terms. - _Washington Bomfim_, Nov 30 2020
%D J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
%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 Andrew Howroyd, <a href="/A000631/b000631.txt">Table of n, a(n) for n = 2..500</a>
%H Washington Bomfim, <a href="/A000631/a000631_4.png">Illustration of initial terms</a>
%H J. L. Faulon, D. Visco and D. Roe, <a href="http://www.cs.sandia.gov/~jfaulon/PUBLICATION/Rev-Comp-Chem.pdf-save">Enumerating Molecules</a>, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
%H H. R. Henze and C. M. Blair, <a href="http://dx.doi.org/10.1021/ja01329a033">The number of structurally isomeric Hydrocarbons of the Ethylene Series</a>, J. Amer. Chem. Soc., 55 (2) (1933), 680-686.
%H H. R. Henze and C. M. Blair, <a href="/A000631/a000631.pdf">The number of structurally isomeric Hydrocarbons of the Ethylene Series</a>, J. Amer. Chem. Soc., 55 (2) (1933), 680-685. (Annotated scanned copy)
%H H. R. Henze and C. M. Blair, <a href="http://dx.doi.org/10.1021/ja01316a050">The number of structural isomers of the more important types of aliphatic compounds</a>, J. Amer. Chem. Soc., 56 (1) (1934), 157-157.
%H C.-W. Lam, <a href="http://dx.doi.org/10.1023/A:1019129510240">A Mathematical Relationship between the Number of Isomers of Alkenes and Alkynes: A Result Established from the Enumeration of Isomers of Alkenes from Alky Biradicals</a>, J. Math. Chem., 23, 421 (1998).
%H R. J. Mathar, <a href="/A002094/a002094_1.pdf">Illustration for graphs up to 6 carbons</a>
%H R. C. Read, <a href="http://dx.doi.org/10.1007/BFb0067377">Some recent results in chemical enumeration</a>, Lect. Notes Math. 303 (1972), 243-259.
%H R. C. Read, <a href="/A002986/a002986.pdf">Some recent results in chemical enumeration</a>, Preprint, circa 1972. (Annotated scanned copy)
%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 p. 28.
%H N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, <a href="https://doi.org/10.1351/pac198855020379">Computer Generation of Isomeric Structures</a>, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F a(n) = b(1)b(n-1) + b(2)b(n-2) + b(3)b(n-3) + ... + b(n/2)(b(n/2) + 1)/2 when n is even or b(1)b(n-1) + b(2)b(n-2) + b(3)b(n-3) + ... + b((n-1)/2)b((n + 1)/2) when n is odd, where b(n) = A000642(n). - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008
%F a(n) = Sum_{k=1..(n-1)/2}( f(k) * f(n-k) ) + [n mod 2 = 0] * ( f(n/2)^2 + f(n/2) ) / 2 where f(n) = A000642(n+1). - _Washington Bomfim_, Nov 29 2020
%F G.f.: (g(x^2) + g(x)^2)/2 where x*g(x) is the g.f. of A000642. - _Andrew Howroyd_, Dec 01 2020
%o (PARI) \\ Here G(n) is A000598 as g.f., h is A000642.
%o seq(n)={my(g=G(n), h=(subst(g, x, x^2) + g^2)/2); Vec(subst(h, x, x^2) + h^2)/2} \\ _Andrew Howroyd_, Dec 01 2020
%Y Cf. A000642, A000598, A027852 (out-degrees of nodes not limited).
%K nonn
%O 2,3
%A _N. J. A. Sloane_
%E More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008
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