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A182100 The number of connected simple labeled graphs with <= n nodes.

%I A182100 #19 Sep 05 2019 02:41:10
%S A182100 1,2,4,11,65,974,31744,2069971,267270041,68629753650,35171000942708,
%T A182100 36024807353574291,73784587576805254665,302228602363365451957806,
%U A182100 2475873310144021668263093216,40564787336902311168400640561099
%N A182100 The number of connected simple labeled graphs with <= n nodes.
%H A182100 G. C. Greubel, <a href="/A182100/b182100.txt">Table of n, a(n) for n = 0..80</a>
%F A182100 a(n) = Sum_{i=0..n} binomial(n,i)*A001187(i).
%F A182100 E.g.f.: exp(x)*A(x) where A(x) is e.g.f. for A001187.
%F A182100 a(n) = A327078(n) + n. - _Gus Wiseman_, Sep 03 2019
%e A182100 From _Gus Wiseman_, Sep 03 2019: (Start)
%e A182100 The a(0) = 1 through a(3) = 11 edge-sets (singletons represent uncovered vertices):
%e A182100   {}  {}     {}       {}
%e A182100       {{1}}  {{1}}    {{1}}
%e A182100              {{2}}    {{2}}
%e A182100              {{1,2}}  {{3}}
%e A182100                       {{1,2}}
%e A182100                       {{1,3}}
%e A182100                       {{2,3}}
%e A182100                       {{1,2},{1,3}}
%e A182100                       {{1,2},{2,3}}
%e A182100                       {{1,3},{2,3}}
%e A182100                       {{1,2},{1,3},{2,3}}
%e A182100 (End)
%t A182100 nn = 15; g = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn}]; Range[0, nn]! CoefficientList[Series[Exp[x] (Log[g] + 1), {x, 0, nn}], x]
%Y A182100 The unlabeled version is A292300(n) + 1.
%Y A182100 Cf. A001187, A006129, A054592, A287689, A327070, A327075, A327078.
%K A182100 nonn
%O A182100 0,2
%A A182100 _Geoffrey Critzer_, Apr 11 2012