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%I M3341 N1344 #42 Jun 16 2022 23:26:39
%S 0,0,0,0,0,1,1,4,8,22,51,136,335,871,2217,5749,14837,38636,100622,
%T 263381,690709,1817544,4793449,12675741,33592349,89223734,237455566,
%U 633176939,1691377956,4525792533,12129365576,32556355947,87508275471,235529797422
%N Number of alkyl benzenes with n carbon atoms: C(n)H(2n-6).
%D N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 71.
%D 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 p. p. 22, Eq. (H).
%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 Vincenzo Librandi, <a href="/A000639/b000639.txt">Table of n, a(n) for n = 1..100</a>
%H I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, et al., <a href="http://arxiv.org/abs/1408.2021">Enumeration of idempotents in diagram semigroups and algebras</a>, arXiv preprint arXiv:1408.2021 [math.GR], 2014.
%H G. Polya, <a href="https://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 (p. 422).
%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 (p. 422). (Annotated scanned copy)
%F G.f.: (x^6/12)*(B(x)^6+4*B(x^2)^3+2*B(x^3)^2+3*B(x)^2*B(x^2)^2+2*B(x^6)), where B = g.f. of A000598.
%e G.f. = x^6 + x^7 + 4*x^8 + 8*x^9 + 22*x^10 + 51*x^11 + 136*x^12 + 335*x^13 + ...
%e a(8)=4 because the unique isomers are 1,2-Dimethylbenzene; 1,3-Dimethylbenzene; 1,4-Dimethylbenzene, 1-Ethylbenzene. All have formula C(8)H(10)
%t m = 100; For[A = 0; i = 0, i <= m, i++, A = Series[1 + x*(A^3/6 + (A /. x -> x^2)*A/2 + (A /. x -> x^3)/3), {x, 0, m+1}] // Normal]; B[x_] = A; (1/12)*(B[x]^6 + 4*B[x^2]^3 + 2*B[x^3]^2 + 3*B[x]^2*B[x^2]^2 + 2*B[x^6]) + O[x]^m // CoefficientList[#, x]& // Join[{0, 0, 0, 0, 0}, #]& (* _Jean-François Alcover_, Oct 12 2011, updated Nov 24 2016 *)
%Y Cf. A000598 (Alkyl radicals).
%K nonn,easy,nice
%O 1,8
%A _N. J. A. Sloane_
%E Better description from Bruce Corrigan (scentman(AT)myfamily.com), Oct 23 2002
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