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A155863 Triangle T(n,k) = n*(n^2 - 1)*binomial(n-2, k-1) for 1 <= k <= n-1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

%I A155863 #11 Sep 08 2022 08:45:41
%S A155863 1,1,1,1,6,1,1,24,24,1,1,60,120,60,1,1,120,360,360,120,1,1,210,840,
%T A155863 1260,840,210,1,1,336,1680,3360,3360,1680,336,1,1,504,3024,7560,10080,
%U A155863 7560,3024,504,1,1,720,5040,15120,25200,25200,15120,5040,720,1,1,990,7920,27720,55440,69300,55440,27720,7920,990,1
%N A155863 Triangle T(n,k) = n*(n^2 - 1)*binomial(n-2, k-1) for 1 <= k <= n-1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.
%H A155863 G. C. Greubel, <a href="/A155863/b155863.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A155863 T(n, k) = coefficients of p(n, x), where p(n, x) = 1 + x^n + x*((d/dx)^3 (x+1)^(n+1)) and T(0, 0) = 1.
%F A155863 From _Franck Maminirina Ramaharo_, Dec 03 2018: (Start)
%F A155863 T(n, k) = (n-1)*n*(n+1)*binomial(n-2, k-1) with T(n, 0) = T(n, n) = 1.
%F A155863 n-th row polynomial is x^n + n*(n^2 - 1)*x*(x+1)^(n-2) + (1 + (-1)^(2^n))/2.
%F A155863 G.f.: 1/(1 - y) + 1/(1 - x*y) + (6*x*y^2)/(1 - y - x*y)^4 - 1.
%F A155863 E.g.f.: exp(y) + exp(x*y) + (3*x*y^2 + (x + x^2)*y^3)*exp((1 + x)*y) - 1. (End)
%F A155863 Sum_{k=0..n} T(n, k) = 2 - [n=0] + 6*A001789(n+1) = 2 - [n=0] + A052771(n+1). - _G. C. Greubel_, Jun 04 2021
%e A155863 Triangle begins:
%e A155863   1;
%e A155863   1,   1;
%e A155863   1,   6,    1;
%e A155863   1,  24,   24,     1;
%e A155863   1,  60,  120,    60,     1;
%e A155863   1, 120,  360,   360,   120,     1;
%e A155863   1, 210,  840,  1260,   840,   210,     1;
%e A155863   1, 336, 1680,  3360,  3360,  1680,   336,     1;
%e A155863   1, 504, 3024,  7560, 10080,  7560,  3024,   504,    1,
%e A155863   1, 720, 5040, 15120, 25200, 25200, 15120,  5040,  720,   1;
%e A155863   1, 990, 7920, 27720, 55440, 69300, 55440, 27720, 7920, 990, 1;
%e A155863   ...
%t A155863 (* First program *)
%t A155863 p[n_, x_]:= p[n, x]= If[n==0, 1, 1 + x^n + x*D[(x+1)^(n+1), {x, 3}]];
%t A155863 Flatten[Table[CoefficientList[p[n,x], x], {n,0,12}]]
%t A155863 (* Second program *)
%t A155863 T[n_, k_]:= If[k==0 || k==n, 1, 6*Binomial[n+1, 3]*Binomial[n-2, k-1]];
%t A155863 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 04 2021 *)
%o A155863 (Maxima) T(n, k):= ratcoef(expand(x^n + n*(n^2 -1)*x*(x+1)^(n-2) + (1 + (-1)^(2^n))/2), x, k)$
%o A155863 create_list(T(n, k), n, 0, 12, k, 0, n); /* _Franck Maminirina Ramaharo_, Dec 03 2018 */
%o A155863 (Magma)
%o A155863 A155863:= func< n,k | k eq 0 or k eq n select 1 else 6*Binomial(n+1, 3)*Binomial(n-2, k-1) >;
%o A155863 [A155863(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 04 2021
%o A155863 (Sage)
%o A155863 def A155863(n,k): return 1 if (k==0 or k==n) else 6*binomial(n+1, 3)*binomial(n-2, k-1)
%o A155863 flatten([[A155863(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 04 2021
%Y A155863 Cf. A001789, A052771, A155864, A155865.
%K A155863 nonn,tabl,easy
%O A155863 0,5
%A A155863 _Roger L. Bagula_, Jan 29 2009
%E A155863 Edited and name clarified by _Franck Maminirina Ramaharo_, Dec 03 2018