OFFSET
0,1
COMMENTS
Old name: Number of homeomorphically irreducible general graphs on 1 labeled node and with n edges.
A homeomorphically irreducible general graph is a graph with multiple edges and loops and without nodes of degree 2.
This sequence is also produced by Wolfram's Rule 253 of Elementary Cellular Automaton as a triangle read by rows giving successive states initiated with a single ON (black) cell. See the Wolfram, Weisstein and Index links below. - Robert Price, Jan 31 2016
Decimal expansion of 91/900. - Elmo R. Oliveira, May 05 2024
REFERENCES
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.
LINKS
V. Jovovic, Generating functions for homeomorphically irreducible general graphs on n labeled nodes.
V. Jovovic, Recurrences for the numbers of homeomorphically irreducible general graphs on m labeled nodes and n edges.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
S. Wolfram, A New Kind of Science.
Index entries for linear recurrences with constant coefficients, signature (1).
FORMULA
G.f.: (x^2 - x + 1)/(1 - x). a(0)=1, a(1)=0; a(n)=1, n > 1.
E.g.f. for homeomorphically irreducible general graphs with n nodes and k edges is (1 + x*y)^(- 1/2)*exp(- x*y/2 + x^2*y^2/4)*Sum_{k >= 0} 1/(1 - x)^binomial(k + 1, 2)*exp(- x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.
E.g.f.: e^x - x. - Paul Barry, May 06 2007
a(n) = 1 - binomial(0,n-1). - Arkadiusz Wesolowski, Feb 10 2012
MAPLE
1, 0, seq(1, n=2..200); # Wesley Ivan Hurt, Apr 12 2017
PROG
(PARI) a(n)=n!=1 \\ Charles R Greathouse IV, Jun 06 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vladeta Jovovic, Apr 03 2001
EXTENSIONS
Definition simplified by N. J. A. Sloane, Sep 26 2023
STATUS
approved