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A327367
Number of labeled simple graphs with n vertices, at least one of which is isolated.
1
0, 1, 1, 4, 23, 256, 5319, 209868, 15912975, 2343052576, 675360194287, 383292136232380, 430038382710483623, 956430459603341708896, 4224538833207707658410103, 37106500399796746894085512140, 648740170822904504303462104598943
OFFSET
0,4
FORMULA
a(n) = A006125(n) - A006129(n).
MAPLE
b:= proc(n) option remember; `if`(n=0, 1,
2^binomial(n, 2)-add(b(k)*binomial(n, k), k=0..n-1))
end:
a:= n-> 2^(n*(n-1)/2)-b(n):
seq(a(n), n=0..17); # Alois P. Heinz, Sep 04 2019
MATHEMATICA
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#!=Range[n]&]], {n, 0, 5}]
PROG
(PARI) b(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2)); \\ A006129
a(n) = 2^(n*(n-1)/2) - b(n); \\ Michel Marcus, Sep 05 2019
CROSSREFS
The unlabeled version is A000088(n - 1).
Labeled graphs with no isolated vertices are A006129.
Covering graphs with at least one endpoint are A327227.
Sequence in context: A123637 A293510 A234595 * A303652 A130890 A219932
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 04 2019
STATUS
approved