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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

%I M0839 N0318 #73 Nov 29 2020 08:18:48

%S 0,1,1,2,3,7,14,32,72,171,405,989,2426,6045,15167,38422,97925,251275,

%T 648061,1679869,4372872,11428365,29972078,78859809,208094977,

%U 550603722,1460457242,3882682803,10344102122,27612603765,73844151259,197818389539

%N a(1)=0; for n>1, a(n) = number of isomeric hydrocarbons of the acetylene series with carbon content n.

%C The former definition was "Number of alkyl derivatives of acetylene X^{II} C_n H_{2n+2} with n carbon atoms" with offset 0.

%C 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

%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 T. D. Noe, <a href="/A000642/b000642.txt">Table of n, a(n) for n = 1..201</a> (offset adapted by _Georg Fischer_, Jan 31 2019)

%H Washington Bomfim, <a href="/A000642/a000642.png">Illustration of initial terms</a>

%H D. D. Coffman, C. M. Blair and H. R. Henze, <a href="https://doi.org/10.1021/ja01328a029">The number of structurally isomeric hydrocarbons of the acetylene series</a>, J. Amer. Chem. Soc., 55 (1933), 252-253.

%H D. D. Coffman, C. M. Blair and H. R. Henze, <a href="/A000642/a000642.pdf">The number of structurally isomeric hydrocarbons of the acetylene series</a>, J. Amer. Chem. Soc., 55 (1933), 252-253. (Annotated scanned copy)

%H Jean-Loup Faulon, Donald P. Visco Jr., Diana Roe, <a href="https://onlinelibrary.wiley.com/doi/abs/10.1002/0471720895.ch3">Enumerating Molecules</a>, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.

%H R. J. Mathar, <a href="/A002094/a002094_1.pdf">Illustration for graphs up to 6 carbons</a>

%H G. Polya, <a href="http://dx.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; line 8 of Table I, "R" of Table IV.

%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; line 8 of Table I, "R" of Table IV. (Annotated scanned copy)

%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="/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 G.f.: A(x)=(1/2)*x*(B(x^2)+B(x)^2), where B(x) = g.f. for A000598.

%F a(n) ~ c * d^n / n^(3/2), where d = 1/A261340 = 2.815460033176... and c = 0.13833565403175156418512996853... - _Vaclav Kotesovec_, Feb 11 2019

%t 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];

%t A[x_] = (1/2)*x*(B[x^2] + B[x]^2) + O[x]^terms;

%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Jun 28 2012, updated Jan 10 2018 *)

%o (PARI) \\ here G(n) is A000598 as g.f.

%o 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}

%o seq(n)={my(g=G(n)); Vec(subst(g,x,x^2) + g^2, -(n+1))/2} \\ _Andrew Howroyd_, Nov 28 2020

%Y Cf. A000598, A002094.

%K nonn,easy,nice

%O 1,4

%A _N. J. A. Sloane_

%E I changed the definition and offset so as to agree with Coffman et al. (1933). - _N. J. A. Sloane_, Jan 13 2019

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