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

3,2

Links

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

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.

Example

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

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 *)

Prog

(PARI) {T(n, k) = 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
(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
(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

Crossrefs

Row sums: A057500, columns: A053507, A065889.

Sequence in context: A130895 A367431 A038329 * A261403 A010202 A079792

Adjacent sequences: A098906 A098907 A098908 * A098910 A098911 A098912

Keyword

easy,nonn,tabl

Author

Vladeta Jovovic, Oct 15 2004

Status

approved