A237195 Number of simple labeled graphs on n nodes that contain some size k connected component, all of whose nodes are labeled with integers {1,2,...,k} for some k in {1,2,...,n}.

1, 2, 7, 52, 846, 28628, 1928768, 255610528, 66822534992, 34632302913632, 35711543058158592, 73426371674544520192, 301419451958411673103360, 2472252535617096234970201088, 40532629372281642451697543062528, 1328660058258732602631909956943781888

Offset: 1

Author

Geoffrey Critzer, Feb 04 2014

Keywords

• nonn

Comments

In other words, a(n) is the number of simple labeled graphs on {1,2,...,n} such that 1 is an isolated node, or 1 and 2 form a size 2 component, or 1,2 and 3 form a size 3 component, or ... 1,2,3,...,k form a size k component, where 1<=k<=n.

Examples

			a(3) = 7. We count all 8 simple labeled graphs on {1,2,3} except: 1-3 2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..80

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
      add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
    end:
a:= n-> add(b(k)*2^((n-k)*(n-k-1)/2), k=1..n):
seq(a(n), n=1..20);  # Alois P. Heinz, Feb 04 2014

Mathematica

nn=15;g=Sum[2^Binomial[n,2]x^n/n!,{n,0,nn}];a=Drop[Range[0,nn]!CoefficientList[Series[Log[g],{x,0,nn}],x],1];Map[Total,Table[Table[Drop[Transpose[Table[ Range[0,nn]!CoefficientList[Series[a[[n]]x^n/n! g,{x,0,nn}],x],{n,1,nn}]],1][[i,j]]/Binomial[i,j],{j,1,i}],{i,1,nn}]]

Formula

a(n) = Sum_{k=1..n} A223894(n,k)/binomial(n,k).