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

%I A098909 #18 Sep 08 2022 08:45:15
%S A098909 1,12,3,150,60,12,2160,1080,360,60,36015,20580,8820,2520,360,688128,
%T A098909 430080,215040,80640,20160,2520,14880348,9920232,5511240,2449440,
%U A098909 816480,181440,20160,360000000,252000000,151200000,75600000,30240000,9072000
%N A098909 Triangle T(n,k) of numbers of connected (unicyclic) graphs with unique cycle of length k (3<=k<=n), on n labeled nodes.
%H A098909 G. C. Greubel, <a href="/A098909/b098909.txt">Rows n = 3..100 of triangle, flattened</a>
%F A098909 T(n, k) = (n-1)!*n^(n-k)/(2*(n-k)!).
%F A098909 E.g.f.: -(2*log(1+x*LambertW(-y))-2*x*LambertW(-y)+x^2*LambertW(-y)^2)/4.
%e A098909 Triangle begins as:
%e A098909       1;
%e A098909      12,     3;
%e A098909     150,    60,   12;
%e A098909    2160,  1080,  360,   60;
%e A098909   36015, 20580, 8820, 2520, 360;
%e A098909   ...
%t A098909 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 A098909 Table[k!*Binomial[n,k]*n^(n-k-1)/2, {n,3,12}, {k,3,n}]//Flatten (* _G. C. Greubel_, May 16 2019 *)
%o A098909 (PARI) {T(n,k) = k!*binomial(n,k)*n^(n-k-1)/2 }; \\ _G. C. Greubel_, May 16 2019
%o A098909 (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 A098909 (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 A098909 (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 A098909 Row sums: A057500, columns: A053507, A065889.
%K A098909 easy,nonn,tabl
%O A098909 3,2
%A A098909 _Vladeta Jovovic_, Oct 15 2004