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%I #14 Jan 14 2020 01:03:43
%S 1,2,3,1,8,3,22,12,116,38,2,854,181,9,11125,1176,45,261083,13351,233,
%T 1,11716594,287048,1513,13,1006700566,12281514,15707,77,164059830598,
%U 1031031446,310050,498,50335907869220,166110813984,12681157,3585,6,29003487462848916,50667148763414,1045586096,37005,57,31397381142761241984,29104659809235092,167233146488,684742,462
%N Triangular array read by rows. T(n,k) is the number of simple unlabeled graphs with n vertices whose components belong to exactly k distinct isomorphism classes.
%F G.f.: Product_{k>=1} (y/(1-x^k) - y + 1)^A001349(k).
%e Triangle begins:
%e 1;
%e 2;
%e 3, 1;
%e 8, 3;
%e 22, 12;
%e 116, 38, 2;
%e 854, 181, 9;
%e 11125, 1176, 45;
%e 261083, 13351, 233, 1;
%e 11716594, 287048, 1513, 13;
%e T(4,2)=3 because we have *-* * * , *-*-* * , a triangle with an isolated point.
%t Needs["Combinatorica`"]; max = 10;
%t A000088 = Table[NumberOfGraphs[n], {n, 0, max}];
%t f[x_] = 1 - Product[1/(1 - x^k)^a[k], {k, 1, max}];
%t a[0] = a[1] = a[2] = 1; coes = CoefficientList[Series[f[x], {x, 0, max}], x];
%t sol = Solve[Thread[Rest[coes + A000088] == 0]];
%t c = Drop[Table[a[n], {n, 0, max}] /. sol // Flatten, 1];
%t Map[Select[#, # > 0 &] &, Drop[CoefficientList[ Series[Product[(y/(1 - x^k) - y + 1)^c[[k]], {k, 1, max}], {x, 0, max}], {x, y}], 1]] // Grid (* after code by Jean-François Alcover in A001349 *)
%Y Cf. A217955.
%K nonn,tabf
%O 1,2
%A _Geoffrey Critzer_, Jan 10 2020
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