A224065 Triangular array read by rows. T(n,k) is the number of size k connected components over all simple unlabeled graphs with n nodes; n>=1,1<=k<=n.
1, 2, 1, 4, 1, 2, 8, 3, 2, 6, 19, 5, 4, 6, 21, 53, 14, 10, 12, 21, 112, 209, 39, 24, 24, 42, 112, 853, 1253, 170, 72, 72, 84, 224, 853, 11117, 13599, 1083, 322, 210, 231, 448, 1706, 11117, 261080, 288267, 12516, 2112, 948, 735, 1232, 3412, 22234, 261080, 11716571
Offset: 1
Examples
1, 2, 1, 4, 1, 2, 8, 3, 2, 6, 19, 5, 4, 6, 21, 53, 14, 10, 12, 21, 112, 209, 39, 24, 24, 42, 112, 853, 1253, 170, 72, 72, 84, 224, 853, 11117, 13599, 1083, 322, 210, 231, 448, 1706, 11117, 261080,
Crossrefs
Cf. A223894 (labeled version).
Programs
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Mathematica
nn=10;h[list_]:=Select[list,#>0&];f[list_]:=Total[Table[list[[i]]*(i-1),{i,1,Length[list]}]];g[x_]:=Sum[NumberOfGraphs[n]x^n,{n,0,nn}];c[x_]:=Sum[a[n]x^n,{n,0,nn}];a[0]=1;sol=SolveAlways[g[x]==Normal[Series[Product[1/(1-x^i)^a[i],{i,1,nn}],{x,0,nn}]],x];b=Drop[Flatten[Table[a[n],{n,0,nn}]/.sol],1];Map[h,Drop[Transpose[Table[Map[f,CoefficientList[Series[(1/(1-y x^n)^b[[n]])Product[1/(1- x^i)^b[[i]],{i,1,nn}](1-x^n)^b[[n]],{x,0,nn}],{x,y}]],{n,1,nn}]],1]]//Flatten
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