A000642 a(1)=0; for n>1, a(n) = number of isomeric hydrocarbons of the acetylene series with carbon content n.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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)
• Washington Bomfim, Illustration of initial terms
• 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.
• R. J. Mathar, Illustration for graphs up to 6 carbons
• 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.
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A000598, A002094.

Programs

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 *)

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

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

Extensions

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