A000639 Number of alkyl benzenes with n carbon atoms: C(n)H(2n-6).
0, 0, 0, 0, 0, 1, 1, 4, 8, 22, 51, 136, 335, 871, 2217, 5749, 14837, 38636, 100622, 263381, 690709, 1817544, 4793449, 12675741, 33592349, 89223734, 237455566, 633176939, 1691377956, 4525792533, 12129365576, 32556355947, 87508275471, 235529797422
Offset: 1
Examples
G.f. = x^6 + x^7 + 4*x^8 + 8*x^9 + 22*x^10 + 51*x^11 + 136*x^12 + 335*x^13 + ... 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)
References
- N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 71.
- 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).
- 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).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..100
- I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, et al., Enumeration of idempotents in diagram semigroups and algebras, arXiv preprint arXiv:1408.2021 [math.GR], 2014.
- G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443 (p. 422).
- G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443 (p. 422). (Annotated scanned copy)
Crossrefs
Cf. A000598 (Alkyl radicals).
Programs
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Mathematica
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 *)
Formula
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.
Extensions
Better description from Bruce Corrigan (scentman(AT)myfamily.com), Oct 23 2002