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A000602
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Number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) ignoring stereoisomers.
(Formerly M0718 N0267)
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76
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1, 1, 1, 1, 2, 3, 5, 9, 18, 35, 75, 159, 355, 802, 1858, 4347, 10359, 24894, 60523, 148284, 366319, 910726, 2278658, 5731580, 14490245, 36797588, 93839412, 240215803, 617105614, 1590507121, 4111846763, 10660307791, 27711253769
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OFFSET
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0,5
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COMMENTS
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Trees are unrooted, nodes are unlabeled. Every node has degree <= 4.
Ignoring stereoisomers means that the children of a node are unordered. They can be permuted in any way and it is still the same tree. See A000628 for the analogous sequence with stereoisomers counted.
In alkanes every carbon has valence exactly 4 and every hydrogen has valence exactly 1. But the trees considered here are just the carbon "skeletons" (with all the hydrogen atoms stripped off) so now each carbon bonds to 1 to 4 other carbons. The degree of each node is then <= 4.
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REFERENCES
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K. Adam, Title?, MNU 1983, 36, 29 (in German).
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 290.
L. Bytautats, D. J. Klein, Alkane Isomere Combinatorics: Stereostructure enumeration and graph-invariant and molecylar-property distributions, J. Chem. Inf. Comput. Sci 39 (1999) 803, Table 1.
A. Cayley, Über die analytischen Figuren, welche in der Mathematik Baeume genannt werden..., Chem. Ber. 8 (1875), 1056-1059.
R. Davies and P. J. Freyd, C_{167}H_{336} is The Smallest Alkane with More Realizable Isomers than the Observable Universe has Particles, Journal of Chemical Education, Vol. 66, 1989, pp. 278-281.
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6.1 Chemical Isomers, p. 299.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 529.
Handbook of Combinatorics, North-Holland '95, p. 1963.
J. B. Hendrickson and C. A. Parks, "Generation and Enumeration of Carbon skeletons", J. Chem. Inf. Comput. Sci, vol. 31 (1991) pp. 101-107. See Table 2, column 2 on page 103.
M. D. Jackson and T. I. Bieber, Applications of degree distribution, 2: construction and enumeration of isomers in the alkane series, J. Chem. Info. and Computer Science, 33 (1993), 701-708.
J. Lederberg et al., Applications of artificial intelligence for chemical systems, I: The number of possible organic compounds. Acyclic structures containing C, H, O and N, J. Amer. Chem. Soc., 91 (1969), 2973-2097.
L. M. Masinter, Applications of artificial intelligence for chemical systems, XX, Exhaustive generation of cyclic and acyclic isomers, J. Amer. Chem. Soc., 96 (1974), 7702-7714.
D. Perry, The number of structural isomers ..., J. Amer. Chem. Soc. 54 (1932), 2918-2920. [Gives a(60) correctly - compare first link below]
M. Petkovsek and T. Pisanski, Counting disconnected structures: chemical trees, fullerenes, I-graphs and others, Croatica Chem. Acta, 78 (2005), 563-567.
D. H. Rouvray, An introduction to the chemical applications of graph theory, Congress. Numerant., 55 (1986), 267-280. - N. J. A. Sloane, Apr 08 2012
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).
Marten J. ten Hoor, Formula for Success?, Education in Chemistry, 2005, 42(1), 10.
S. Wagner, Graph-theoretical enumeration and digital expansions: an analytic approach, Dissertation, Fakult. f. Tech. Math. u. Tech. Physik, Tech. Univ. Graz, Austria, Feb., 2006.
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LINKS
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F. Hermann, Entgegnung, Chem. Ber., 31 (1898), 91. [III (1898), Annotated scanned copy]
J. Lederberg, The topology of molecules, in The Mathematical Sciences. Cambridge, Mass.: MIT Press, 1969, pages 37-51. [Annotated scanned copy]
J. Lederberg, Dendral-64, I, Report to NASA, Dec 1964 [Annotated scanned copy]
J. Lederberg, Dendral-64, II, Report to NASA, Dec 1965 [Annotated scanned copy]
J. Lederberg, Dendral-64, III, Report to NASA, Dec 1968 [Annotated scanned copy]
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FORMULA
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Also equals A000022 + A000200 (n>0), both of which have known generating functions. Also g.f. = A000678(x) - A000599(x) + A000598(x^2) = (x + x^2 + 2x^3 + ...) - (x^2 + x^3 + 3x^4 + ...) + (1 + x^2 + x^4 + ...) = 1 + x + x^2 + x^3 + 2x^4 + 3x^5 + ...
G.f.: B(x) - cycle_index(S2,-B(x)) + x * cycle_index(S4,B(x)) = B(x) - (B(x)^2 - B(x^2)) / 2 + x * (B(x)^4 + 6*B(x)^2*B(x^2) + 8*B(x)*B(x^3) + 3*B(x^2)^2 + 6*B(x^4)) / 24, where B(x) = 1 + x * cycle_index(S3,B(x)) = 1 + x * (B(x)^3 + 3*B(x)*B(x^2) + 2*B(x^3)) / 6 is the generating function for A000598. - Robert A. Russell, Jan 16 2023
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EXAMPLE
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a(6)=5 because hexane has five isomers: n-hexane; 2-methylpentane; 3-methylpentane; 2,2-dimethylbutane; 2,3-dimethylbutane. - Michael Lugo (mtlugo(AT)mit.edu), Mar 15 2003 (corrected by Andrey V. Kulsha, Sep 22 2011)
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MAPLE
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if n=0 then
1
else
end if;
end proc:
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MATHEMATICA
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n = 40; (* algorithm from Rains and Sloane *)
S3[f_, h_, x_] := f[h, x]^3/6 + f[h, x] f[h, x^2]/2 + f[h, x^3]/3;
S4[f_, h_, x_] := f[h, x]^4/24 + f[h, x]^2 f[h, x^2]/4 + f[h, x] f[h, x^3]/3 + f[h, x^2]^2/8 + f[h, x^4]/4;
T[-1, z_] := 1; T[h_, z_] := T[h, z] = Table[z^k, {k, 0, n}].Take[CoefficientList[z^(n+1) + 1 + S3[T, h-1, z]z, z], n+1];
Sum[Take[CoefficientList[z^(n+1) + S4[T, h-1, z]z - S4[T, h-2, z]z - (T[h-1, z] - T[h-2, z]) (T[h-1, z]-1), z], n+1], {h, 1, n/2}] + PadRight[{1, 1}, n+1] + Sum[Take[CoefficientList[z^(n+1) + (T[h, z] - T[h-1, z])^2/2 + (T[h, z^2] - T[h-1, z^2])/2, z], n+1], {h, 0, n/2}] (* Robert A. Russell, Sep 15 2018 *)
b[n_, i_, t_, k_] := b[n, i, t, k] = If[i<1, 0, Sum[Binomial[b[i-1, i-1,
k, k] + j-1, j]* b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]];
b[0, i_, t_, k_] = 1; m = 3; (* m = maximum children *) n = 40;
gf[x_] = 1 + Sum[b[j-1, j-1, m, m]x^j, {j, 1, n}]; (* G.f. for A000598 *)
ci[x_] = SymmetricGroupIndex[m+1, x] /. x[i_] -> gf[x^i];
CoefficientList[Normal[Series[gf[x] - (gf[x]^2 - gf[x^2])/2 + x ci[x],
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CROSSREFS
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KEYWORD
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nonn,easy,core,nice
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AUTHOR
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EXTENSIONS
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Additional comments from Steve Strand (snstrand(AT)comcast.net), Aug 20 2003
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STATUS
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approved
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