|
%I #12 Sep 05 2019 02:30:18
%S 0,1,1,4,23,256,5319,209868,15912975,2343052576,675360194287,
%T 383292136232380,430038382710483623,956430459603341708896,
%U 4224538833207707658410103,37106500399796746894085512140,648740170822904504303462104598943
%N Number of labeled simple graphs with n vertices, at least one of which is isolated.
%F a(n) = A006125(n) - A006129(n).
%p b:= proc(n) option remember; `if`(n=0, 1,
%p 2^binomial(n, 2)-add(b(k)*binomial(n, k), k=0..n-1))
%p end:
%p a:= n-> 2^(n*(n-1)/2)-b(n):
%p seq(a(n), n=0..17); # _Alois P. Heinz_, Sep 04 2019
%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#!=Range[n]&]],{n,0,5}]
%o (PARI) b(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2)); \\ A006129
%o a(n) = 2^(n*(n-1)/2) - b(n); \\ _Michel Marcus_, Sep 05 2019
%Y The unlabeled version is A000088(n - 1).
%Y Labeled graphs with no isolated vertices are A006129.
%Y Covering graphs with at least one endpoint are A327227.
%Y Cf. A006125, A006129, A054592, A245797, A327103, A327105.
%K nonn
%O 0,4
%A _Gus Wiseman_, Sep 04 2019
| 