|
|
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
|
Jean-Loup Faulon, Donald P. Visco Jr., Diana Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
|
|
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;
|
|
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
|
|
|
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
|
|
|
|