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A060576 Number of homeomorphically irreducible general graphs on 1 labeled node and with n edges. 20
1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

REFERENCES

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

LINKS

Table of n, a(n) for n=0..104.

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 to Elementary Cellular Automata

Index entries for sequences related to cellular automata

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 - C(1,n) + C(0,n), with n>=0. - Paolo P. Lava, Feb 15 2008

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

Cf. A003514, A060516, A060533-A060537, A060577-A060581.

Sequence in context: A324113 A105812 A134323 * A261012 A019590 A154955

Adjacent sequences:  A060573 A060574 A060575 * A060577 A060578 A060579

KEYWORD

nonn,easy

AUTHOR

Vladeta Jovovic, Apr 03 2001

STATUS

approved

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Last modified December 6 01:25 EST 2021. Contains 349558 sequences. (Running on oeis4.)