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A217763
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.
0
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
OFFSET
3,2
COMMENTS
Column k=0 is A001205.
Row sums are A137916.
FORMULA
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).
EXAMPLE
....o-o..........o-o......
....| |..........|\ ......
....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.
MATHEMATICA
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
CROSSREFS
Sequence in context: A263672 A233287 A214528 * A234749 A024546 A073542
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Mar 23 2013
STATUS
approved