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A000642 a(1)=0; for n>1, a(n) = number of isomeric hydrocarbons of the acetylene series with carbon content n.
(Formerly M0839 N0318)
12
0, 1, 1, 2, 3, 7, 14, 32, 72, 171, 405, 989, 2426, 6045, 15167, 38422, 97925, 251275, 648061, 1679869, 4372872, 11428365, 29972078, 78859809, 208094977, 550603722, 1460457242, 3882682803, 10344102122, 27612603765, 73844151259, 197818389539 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
The former definition was "Number of alkyl derivatives of acetylene X^{II} C_n H_{2n+2} with n carbon atoms" with offset 0.
a(n+1) is the number of rooted trees with n nodes and out-degree <= 2 on the root and out-degree <= 3 on all other nodes. See illustration of initial terms. - Washington Bomfim, Nov 28 2020
REFERENCES
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
T. D. Noe, Table of n, a(n) for n = 1..201 (offset adapted by Georg Fischer, Jan 31 2019)
D. D. Coffman, C. M. Blair and H. R. Henze, The number of structurally isomeric hydrocarbons of the acetylene series, J. Amer. Chem. Soc., 55 (1933), 252-253.
D. D. Coffman, C. M. Blair and H. R. Henze, The number of structurally isomeric hydrocarbons of the acetylene series, J. Amer. Chem. Soc., 55 (1933), 252-253. (Annotated scanned copy)
Jean-Loup Faulon, Donald P. Visco Jr., Diana Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; line 8 of Table I, "R" of Table IV.
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; line 8 of Table I, "R" of Table IV. (Annotated scanned copy)
R. C. Read, Some recent results in chemical enumeration, Lect. Notes Math. 303 (1972), 243-259.
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. [Annotated scanned copy] See p. 28.
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
FORMULA
G.f.: A(x)=(1/2)*x*(B(x^2)+B(x)^2), where B(x) = g.f. for A000598.
a(n) ~ c * d^n / n^(3/2), where d = 1/A261340 = 2.815460033176... and c = 0.13833565403175156418512996853... - Vaclav Kotesovec, Feb 11 2019
MATHEMATICA
terms = 32; B[_] = 0; Do[B[x_] = 1 + (1/6)*x*(B[x]^3 + 3*B[x]*B[x^2] + 2*B[x^3]) + O[x]^terms // Normal, terms];
A[x_] = (1/2)*x*(B[x^2] + B[x]^2) + O[x]^terms;
CoefficientList[A[x], x] (* Jean-François Alcover, Jun 28 2012, updated Jan 10 2018 *)
PROG
(PARI) \\ here G(n) is A000598 as g.f.
G(n)={my(g=O(x)); for(n=1, n, g = 1 + x*(g^3/6 + subst(g, x, x^2)*g/2 + subst(g, x, x^3)/3) + O(x^n)); g}
seq(n)={my(g=G(n)); Vec(subst(g, x, x^2) + g^2, -(n+1))/2} \\ Andrew Howroyd, Nov 28 2020
CROSSREFS
Sequence in context: A205484 A151530 A180752 * A348530 A229734 A035083
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
I changed the definition and offset so as to agree with Coffman et al. (1933). - N. J. A. Sloane, Jan 13 2019
STATUS
approved

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Last modified February 21 13:52 EST 2024. Contains 370235 sequences. (Running on oeis4.)