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.

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

Author

Geoffrey Critzer, Mar 23 2013

Keywords

• nonn
• tabl

Comments

Column k=0 is A001205.

Row sums are A137916.

Examples

			  ....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.
		

Programs

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

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).