A224510 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.

0, 0, 1, 5, 48, 874, 30264, 2019680, 263757552, 68148453616, 35042313517056, 35957170070748800, 73714223732206510848, 302083108644327384484864, 2475273899774743284992553984, 40559859846438312840086623738880, 1329146799084147159829387611140308992

Offset: 0

Author

Geoffrey Critzer, Apr 08 2013

Keywords

• nonn

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..80

Programs

Maple

b:= proc(n) b(n):= `if`(n=0, 1, 2^binomial(n, 2)-
      add(binomial(n, k)*k*b(k)*2^binomial(n-k, 2), k=0..n-1)/n)
    end:
a:= n-> add(binomial(n-2, k)*b(k+2)*2^binomial(n-k-2,2), k=0..n-2):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 09 2013

Mathematica

(* 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}]
(* or *)
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]

Formula

a(n) = Sum_{k=0..n-2} C(n-2,k)*A001187(k+2)*A006125(n-k-2). - Alois P. Heinz, Apr 09 2013

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

Extensions

More terms from Alois P. Heinz, Apr 09 2013