A098909 Triangle T(n,k) of numbers of connected (unicyclic) graphs with unique cycle of length k (3<=k<=n), on n labeled nodes.

1, 12, 3, 150, 60, 12, 2160, 1080, 360, 60, 36015, 20580, 8820, 2520, 360, 688128, 430080, 215040, 80640, 20160, 2520, 14880348, 9920232, 5511240, 2449440, 816480, 181440, 20160, 360000000, 252000000, 151200000, 75600000, 30240000, 9072000

Offset: 3

Author

Vladeta Jovovic, Oct 15 2004

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins as:
      1;
     12,     3;
    150,    60,   12;
   2160,  1080,  360,   60;
  36015, 20580, 8820, 2520, 360;
  ...
		

Links

• G. C. Greubel, Rows n = 3..100 of triangle, flattened

Crossrefs

Row sums: A057500, columns: A053507, A065889.

Programs

GAP

Flat(List([3..12], n-> List([3..n], k-> Factorial(k)*Binomial(n,k) *n^(n-k-1)/2 ))); # G. C. Greubel, May 16 2019

Magma

[[Factorial(k)*Binomial(n,k)*n^(n-k-1)/2: k in [3..n]]: n in [3..12]]; // G. C. Greubel, May 16 2019

Mathematica

f[list_] := Select[list, #>0&]; t = Sum[n^(n-1)x^n/n!, {n, 1, 20}]; Map[f,Drop[Transpose[Table[Range[0,8]! CoefficientList[Series[t^n/(2n), {x, 0, 8}], x], {n, 3, 8}]], 3]] (* Geoffrey Critzer, Oct 23 2011 *)
Table[k!*Binomial[n,k]*n^(n-k-1)/2, {n,3,12}, {k,3,n}]//Flatten (* G. C. Greubel, May 16 2019 *)

PARI

{T(n,k) = k!*binomial(n,k)*n^(n-k-1)/2 }; \\ G. C. Greubel, May 16 2019

Sage

[[factorial(k)*binomial(n,k)*n^(n-k-1)/2 for k in (3..n)] for n in (3..12)] # G. C. Greubel, May 16 2019

Formula

T(n, k) = (n-1)!*n^(n-k)/(2*(n-k)!).

E.g.f.: -(2*log(1+x*LambertW(-y))-2*x*LambertW(-y)+x^2*LambertW(-y)^2)/4.