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A201922
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Triangle read by rows: T(n,m) = number of unlabeled graphs on n nodes with m connected components, m = 1,2,...,n.
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6
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1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 21, 8, 3, 1, 1, 112, 30, 9, 3, 1, 1, 853, 145, 32, 9, 3, 1, 1, 11117, 1028, 154, 33, 9, 3, 1, 1, 261080, 12320, 1065, 156, 33, 9, 3, 1, 1, 11716571, 274806, 12513, 1074, 157, 33, 9, 3, 1, 1, 1006700565, 12007355, 276114, 12550, 1076, 157, 33, 9, 3, 1, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,4
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LINKS
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Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 5 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
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FORMULA
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T(n,m) = sum over the partitions of n with m parts: 1*K1 + 2*K2 + ... + n*Kn = n, K1 + K2 + ... + Kn = m, of Product_{i=1..n} binomial(A001349(i) + Ki - 1, Ki).
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EXAMPLE
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Triangle starts:
1
1 1
2 1 1
6 3 1 1
21 8 3 1 1
112 30 9 3 1 1
853 145 32 9 3 1 1 ...
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MATHEMATICA
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nn=10; 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[1/(1-y x^n)^c[[n+1]], {n, 1, nn}]; Map[f, Drop[CoefficientList[Series[g, {x, 0, nn}], {x, y}], 1]] //Flatten (* Geoffrey Critzer, Apr 19 2012 (c in above Mma code is given by Jean Francois Alcover in A001349) *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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