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

3,2

Comments

Column k=0 is A001205.

Row sums are A137916.

Links

Table of n, a(n) for n=3..38.

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

Adjacent sequences: A217760 A217761 A217762 * A217764 A217765 A217766

Keyword

nonn,tabl

Author

Geoffrey Critzer, Mar 23 2013

Status

approved