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A331123
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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.
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0
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1, 2, 3, 1, 8, 3, 22, 12, 116, 38, 2, 854, 181, 9, 11125, 1176, 45, 261083, 13351, 233, 1, 11716594, 287048, 1513, 13, 1006700566, 12281514, 15707, 77, 164059830598, 1031031446, 310050, 498, 50335907869220, 166110813984, 12681157, 3585, 6, 29003487462848916, 50667148763414, 1045586096, 37005, 57, 31397381142761241984, 29104659809235092, 167233146488, 684742, 462
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f.: Product_{k>=1} (y/(1-x^k) - y + 1)^A001349(k).
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EXAMPLE
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Triangle begins:
1;
2;
3, 1;
8, 3;
22, 12;
116, 38, 2;
854, 181, 9;
11125, 1176, 45;
261083, 13351, 233, 1;
11716594, 287048, 1513, 13;
T(4,2)=3 because we have *-* * * , *-*-* * , a triangle with an isolated point.
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MATHEMATICA
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Needs["Combinatorica`"]; max = 10;
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 = Solve[Thread[Rest[coes + A000088] == 0]];
c = Drop[Table[a[n], {n, 0, max}] /. sol // Flatten, 1];
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 *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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