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A370165
Number of labeled loop-graphs covering n vertices without a non-loop edge with loops at both ends.
2
1, 1, 4, 29, 400, 10289, 496548, 45455677, 7983420736, 2716094133313, 1803251169342820, 2348787270663723581, 6024912118926389490448, 30516957491540079828757553, 305811332460677494410532494660, 6071677788061208810793717466942237
OFFSET
0,3
COMMENTS
Number of ways to choose a stable vertex set of a simple graph with n vertices.
LINKS
FORMULA
Inverse binomial transform of A079491.
E.g.f.: Sum_{k >= 0} exp((2^k-1)*x)*2^(k*(k-1)/2)*x^k/k!. - Andrew Howroyd, Feb 20 2024
EXAMPLE
The a(3) = 29 loop-graphs (loops shown as singletons):
{1,23} {1,2,3} {1,2,13,23}
{2,13} {1,2,13} {1,3,12,23}
{3,12} {1,2,23} {2,3,12,13}
{12,13} {1,3,12} {1,12,13,23}
{12,23} {1,3,23} {2,12,13,23}
{13,23} {2,3,12} {3,12,13,23}
{2,3,13}
{1,12,13}
{1,12,23}
{1,13,23}
{2,12,13}
{2,12,23}
{2,13,23}
{3,12,13}
{3,12,23}
{3,13,23}
{12,13,23}
MATHEMATICA
Table[Length[Select[Subsets[Subsets[Range[n], {1, 2}]], Union@@#==Range[n]&&!MatchQ[#, {___, {x_}, ___, {y_}, ___, {x_, y_}, ___}]&]], {n, 0, 5}]
PROG
(PARI) seq(n)={Vec(serlaplace(sum(k=0, n, exp((2^k-1)*x + O(x*x^n))*2^(k*(k-1)/2)*x^k/k!)))} \\ Andrew Howroyd, Feb 20 2024
CROSSREFS
Without loops we have A006129, connected A001187.
The non-covering version is A079491.
The unlabeled version is A370166, non-covering A339832.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 counts labeled loop-graphs (shifted left), covering A322661.
Sequence in context: A135485 A210526 A221079 * A162287 A324227 A277357
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 12 2024
STATUS
approved