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A125207 Total number of connected components in all subgraphs obtained from the complete labeled graph K_n by removing zero or more edges.

%I A125207 #23 Feb 02 2024 18:31:56
%S A125207 1,3,13,98,1398,39956,2354240,286394544,71225744048,35884971729760,
%T A125207 36419817759267072,74221711070826087424,303193538300703211111936,
%U A125207 2480118087478081928075065344,40601989279034990139321984265216,1329877330680067685563700135615633408
%N A125207 Total number of connected components in all subgraphs obtained from the complete labeled graph K_n by removing zero or more edges.
%C A125207 a(n)/A006125(n) is the expected number of connected components in a simple labeled graph on n vertices. - _Geoffrey Critzer_, May 09 2011
%H A125207 Alois P. Heinz, <a href="/A125207/b125207.txt">Table of n, a(n) for n = 1..82</a>
%F A125207 E.g.f.: (F(x)-1)*exp(F(x)-1) = G(x)*log(G(x)) where G(x) = Sum_{n>=0} 2^(n(n-1)/2) * x^n/n! and F(x) = 1+log(G(x)) is the e.g.f. of A001187.
%F A125207 a(n) = Sum_{k=1..n} k * A143543(n,k). - _Alois P. Heinz_, Feb 02 2024
%e A125207 For n=2, we have two graph on two vertices: complete and empty, the former has one connected component while the latter has two connected components. The total number of connected components is 3, which is a(2).
%t A125207 f[list_]:= Total[Table[i list[[i]],{i,1,Length[list]}]];
%t A125207 a= Sum[2^Binomial[n,2] x^n/n!,{n,0,20}];
%t A125207 Map[f, Transpose[Table[Rest[Range[0, 20]! CoefficientList[Series[Log[a]^k/k!, {x, 0, 20}],x]], {k, 1, 20}]]] (* _Geoffrey Critzer_, May 09 2011 *)
%o A125207 (PARI) G=sum(n=0,30,2^(n*(n-1)/2)*x^n/n!) + O(x^31); v=Vec(G*log(G)); for(i=1,length(v),v[i]*=i!); print(v)
%Y A125207 Cf. A001187, A125205, A125206, A143543.
%K A125207 nonn
%O A125207 1,2
%A A125207 _Max Alekseyev_, Nov 23 2006