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%I #21 Sep 07 2019 08:50:18
%S 1,1,2,3,1,8,3,22,12,117,37,2,854,182,8,11140,1163,43,261085,13365,
%T 218,11716804,286878,1474,12,1006700566,12281795,15449,54,
%U 164059836867,1031025763,309546,416,50335907869220,166110822083,12673543,3106,29003487463212294,50667148427178,1045561143,34873,31397381142761243738,29104659809891176,167232513148,660454,252
%N Triangular array read by rows. T(n,k) is the number of simple unlabeled graphs with n nodes having exactly k distinct components.
%C These graphs may contain identical components but they have a total of k different "types". Cf. A207828.
%C Row sums = A000088.
%F O.g.f.: Product_{n>=1}: y/(1-x^n)^A001349(n) - y + 1, where A001349 is the number of connected graphs.
%e 1
%e 1
%e 2
%e 3 1
%e 8 3
%e 22 12
%e 117 37 2
%e 854 182 8
%e T(4,1) = 8 because we have 6 connected graphs and *-* *-*, and * * * * .
%t nn = 15; c = (A000088 = Table[NumberOfGraphs[n], {n, 0, nn}]; f[x_] = 1 - Product[1/(1 - x^k)^a[k], {k, 1, nn}]; a[0] = a[1] = a[2] = 1; coes = CoefficientList[Series[f[x], {x, 0, nn}], x]; sol = First[Solve[Thread[Rest[coes + A000088] == 0]]]; Table[a[n], {n, 0, nn}] /. sol); f[list_] := Select[list, # > 0 &]; g = Product[y/(1 - x^n)^c[[n + 1]] - y + 1, {n, 1, nn}]; Map[f, CoefficientList[Series[g, {x, 0, nn}], {x, y}]] // Flatten (* Mma code for c in the above is given by Jean-Francois Alcover in A001349 *)
%K nonn,tabf
%O 0,3
%A _Geoffrey Critzer_, Apr 19 2012
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