%I #22 Apr 10 2013 01:47:08

%S 0,0,1,5,48,874,30264,2019680,263757552,68148453616,35042313517056,

%T 35957170070748800,73714223732206510848,302083108644327384484864,

%U 2475273899774743284992553984,40559859846438312840086623738880,1329146799084147159829387611140308992

%N Number of simple labeled graphs on {1,2,...,n} such that the node labeled with 1 is in the same component as the node labeled with 2.

%C A001187(n) < a(n) < A006125(n) for n>2.

%H Alois P. Heinz, <a href="/A224510/b224510.txt">Table of n, a(n) for n = 0..80</a>

%F a(n) = Sum_{k=0..n-2} C(n-2,k)*A001187(k+2)*A006125(n-k-2). - _Alois P. Heinz_, Apr 09 2013

%F E.g.f.: Double integral of A''(x)*B(x) dx^2 where A(x) is e.g.f. for A001187 and B(x) is e.g.f. for A006125. - _Geoffrey Critzer_, Apr 09 2013

%p b:= proc(n) b(n):= `if`(n=0, 1, 2^binomial(n, 2)-

%p add(binomial(n, k)*k*b(k)*2^binomial(n-k, 2), k=0..n-1)/n)

%p end:

%p a:= n-> add(binomial(n-2, k)*b(k+2)*2^binomial(n-k-2,2), k=0..n-2):

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Apr 09 2013

%t (* by brute force counting *) nn=10;g=Sum[2^Binomial[n,2]x^n/n!,{n,0,nn}];a=Drop[Range[0,nn]!CoefficientList[Series[Log[g]+1,{x,0,nn}],x],1];f[list_]:=Product[a[[i]],{i,list}];Table[Total[Map[f,Map[Length,Select[SetPartitions[n],MemberQ[#[[1]],2]&],{2}]]],{n,2,nn}]

%t (* or *)

%t nn=30; g=Sum[2^Binomial[n,2]x^n/n!,{n,0,nn+2}];Range[0,nn]!CoefficientList[Series[D[D[Log[g]+1,x],x] g,{x,0,nn}],x]

%K nonn

%O 0,4

%A _Geoffrey Critzer_, Apr 08 2013

%E More terms from _Alois P. Heinz_, Apr 09 2013