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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.

%I A224065 #14 Sep 18 2013 04:01:14
%S A224065 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,
%T A224065 112,853,1253,170,72,72,84,224,853,11117,13599,1083,322,210,231,448,
%U A224065 1706,11117,261080,288267,12516,2112,948,735,1232,3412,22234,261080,11716571
%N 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.
%C A224065 Row sums are A224031.
%C A224065 Column 1 is A006897.
%C A224065 T(n,n) is A001349.
%F A224065 O.g.f. for column k is the derivative with respect to y then evaluated at y = 1 of (1/(1 - y*x^k))^A001349(k) * (1 - x^k)^A001349(k) * Product_{k>=1}1/(1 - x^k)^A001349(k).
%e A224065 1,
%e A224065 2,     1,
%e A224065 4,     1,    2,
%e A224065 8,     3,    2,   6,
%e A224065 19,    5,    4,   6,   21,
%e A224065 53,    14,   10,  12,  21,  112,
%e A224065 209,   39,   24,  24,  42,  112, 853,
%e A224065 1253,  170,  72,  72,  84,  224, 853, 11117,
%e A224065 13599, 1083, 322, 210, 231, 448, 1706, 11117, 261080,
%t A224065 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
%Y A224065 Cf. A223894 (labeled version).
%K A224065 nonn,tabl
%O A224065 1,2
%A A224065 _Geoffrey Critzer_, Mar 30 2013