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A217763
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Triangular array read by rows: T(n,k) is the number of simple labeled graphs on n nodes with unicyclic components having exactly k nodes with degree 1; n>=3, 0<=k<=n-3.
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0
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1, 3, 12, 12, 90, 120, 70, 600, 1800, 1200, 465, 4725, 19530, 31500, 12600, 3507, 42168, 211680, 529200, 529200, 141120, 30016, 414288, 2451456, 7902720, 13124160, 8890560, 1693440, 286884, 4460760, 30413880, 117573120, 266716800, 312439680, 152409600, 21772800
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OFFSET
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3,2
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COMMENTS
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LINKS
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FORMULA
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exp(A(B(x,y)), where A(x) is e.g.f. for A137916 and B(x,y) is e.g.f. for A055302, gives T(n,n-k) (offset).
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EXAMPLE
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....o-o..........o-o......
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....o-o..........o-o......
T(4,0)=3 because the graph on the left has 4 nodes and 0 nodes with degree 1. It has 3 labelings.
T(4,1)=12 because the graph on the right has 4 nodes and 1 node with degree 1. It has 12 labelings.
1,
3, 12,
12, 90, 120,
70, 600, 1800, 1200,
465, 4725, 19530, 31500, 12600,
3507, 42168, 211680, 529200, 529200, 141120,
30016, 414288, 2451456, 7902720, 13124160, 8890560, 1693440.
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MATHEMATICA
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nn=10; f[list_]:=Select[list, #>0&]; t=Sum[Sum[n!/k! StirlingS2[n-1, n-k]y^k x^n/n!, {k, 1, n}], {n, 0, nn}]; Map[Reverse, Map[f, Drop[Range[0, nn]!CoefficientList[Series[ Exp[Log[1/(1-t)]/2-t/2-t^2/4], {x, 0, nn}], {x, y}], 3]]]//Grid
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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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