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%I #20 Oct 11 2017 05:09:44
%S 1,1,2,1,6,2,21,8,112,28,2,853,145,7,11117,1022,34,261080,12320,181,1,
%T 11716571,274785,1266,12,1006700565,12007355,14106,63,164059830476,
%U 1019030127,293756,407,50335907869219,165091859656,12362198,3023,6,29003487462848061,50502058491413,1032671168,33035,51,31397381142761241960,29054157815353374,166176421788,645086,399,63969560113225176176277,31426486309136268658,50672459139597,25830118,3113
%N Triangular array read by rows. T(n,k) is the number of unlabeled graphs on n nodes that have exactly k distinct components (n >= 1).
%C Row sums are A207828.
%C Column 1 is A001349.
%C Column 2 is A216785.
%C Column 3 is A058915.
%F O.g.f.: Product_{n>=1} (1 + y*x^n)^A001349(n).
%e Triangle begins
%e 1;
%e 1;
%e 2, 1;
%e 6, 2;
%e 21, 8;
%e 112, 28, 2;
%e 853, 145, 7;
%e 11117, 1022, 34;
%e 261080, 12320, 181, 1;
%e 11716571, 274785, 1266, 12;
%t Needs["Combinatorica`"];max=20;A000088=Table[NumberOfGraphs[n],{n,0,max}];f[x_]=1-Product[1/(1-x^k)^a[k],{k,1,max}];a[0]=a[1]=a[2]=1;coes=CoefficientList[Series[f[x],{x,0,max}],x];sol=First[Solve[Thread[Rest[coes+A000088]== 0]]];cg=Table[a[n],{n,1,max}]/.sol;CoefficientList[Series[Product[(1+y x^i)^cg[[i]],{i,1,max}],{x,0,max}],{x,y}]//Grid (* after code by _Jean-François Alcover_ in A001349 *)
%K nonn,tabf
%O 1,3
%A _Geoffrey Critzer_, Oct 15 2012
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