%I #18 Sep 08 2022 08:45:15

%S 1,12,3,150,60,12,2160,1080,360,60,36015,20580,8820,2520,360,688128,

%T 430080,215040,80640,20160,2520,14880348,9920232,5511240,2449440,

%U 816480,181440,20160,360000000,252000000,151200000,75600000,30240000,9072000

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

%H G. C. Greubel, <a href="/A098909/b098909.txt">Rows n = 3..100 of triangle, flattened</a>

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

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

%e Triangle begins as:

%e 1;

%e 12, 3;

%e 150, 60, 12;

%e 2160, 1080, 360, 60;

%e 36015, 20580, 8820, 2520, 360;

%e ...

%t 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 *)

%t Table[k!*Binomial[n,k]*n^(n-k-1)/2, {n,3,12}, {k,3,n}]//Flatten (* _G. C. Greubel_, May 16 2019 *)

%o (PARI) {T(n,k) = k!*binomial(n,k)*n^(n-k-1)/2 }; \\ _G. C. Greubel_, May 16 2019

%o (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

%o (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

%o (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

%Y Row sums: A057500, columns: A053507, A065889.

%K easy,nonn,tabl

%O 3,2

%A _Vladeta Jovovic_, Oct 15 2004