A217763 - OEIS

%I #15 Apr 12 2013 12:35:12

%S 1,3,12,12,90,120,70,600,1800,1200,465,4725,19530,31500,12600,3507,

%T 42168,211680,529200,529200,141120,30016,414288,2451456,7902720,

%U 13124160,8890560,1693440,286884,4460760,30413880,117573120,266716800,312439680,152409600,21772800

%N 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.

%C Column k=0 is A001205.

%C   Row sums are A137916.

%F 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).

%e   ....o-o..........o-o......

%e   ....| |..........|\ ......

%e   ....o-o..........o-o......

%e   T(4,0)=3 because the graph on the left has 4 nodes and 0 nodes with degree 1. It has 3 labelings.

%e   T(4,1)=12 because the graph on the right has 4 nodes and 1 node with degree 1.  It has 12 labelings.

%e 1,

%e 3,     12,

%e 12,    90,     120,

%e 70,    600,    1800,    1200,

%e 465,   4725,   19530,   31500,   12600,

%e 3507,  42168,  211680,  529200,  529200,   141120,

%e 30016, 414288, 2451456, 7902720, 13124160, 8890560, 1693440.

%t 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

%K nonn,tabl

%O 3,2

%A _Geoffrey Critzer_, Mar 23 2013