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A155864 Triangle T(n,k) = n*(n-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 A155864 #11 Sep 08 2022 08:45:41
%S A155864 1,1,1,1,2,1,1,6,6,1,1,12,24,12,1,1,20,60,60,20,1,1,30,120,180,120,30,
%T A155864 1,1,42,210,420,420,210,42,1,1,56,336,840,1120,840,336,56,1,1,72,504,
%U A155864 1512,2520,2520,1512,504,72,1,1,90,720,2520,5040,6300,5040,2520,720,90,1
%N A155864 Triangle T(n,k) = n*(n-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 A155864 G. C. Greubel, <a href="/A155864/b155864.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A155864 T(n, k) = coefficients of p(n, x), where p(n, x) = 1 + x^n + x*((d/dx)^2 (1+x)^n), with T(0, 0) = 1.
%F A155864 From _Franck Maminirina Ramaharo_, Dec 04 2018: (Start)
%F A155864 T(n, k) = n*(n-1)*binomial(n-2, k-1) with T(n, 0) = T(n, n) = 1.
%F A155864 n-th row polynomial is x^n + n*(n - 1)*x*(x + 1)^(n - 2) + (1 + (-1)^(2^n))/2.
%F A155864 G.f.: 1/(1 - y) + 1/(1 - x*y) + 2*x*y^2/(1 - y - x*y)^3 - 1.
%F A155864 E.g.f.: exp(y) + exp(x*y) + x*y^2*exp(y + x*y) - 1. (End)
%F A155864 Sum_{k=0..n} T(n, k) = 2 - [n=0] + A001815(n). - _G. C. Greubel_, Jun 04 2021
%e A155864 Triangle begins:
%e A155864   1;
%e A155864   1,  1;
%e A155864   1,  2,   1;
%e A155864   1,  6,   6,    1;
%e A155864   1, 12,  24,   12,    1;
%e A155864   1, 20,  60,   60,   20,    1;
%e A155864   1, 30, 120,  180,  120,   30,    1;
%e A155864   1, 42, 210,  420,  420,  210,   42,    1;
%e A155864   1, 56, 336,  840, 1120,  840,  336,   56,   1;
%e A155864   1, 72, 504, 1512, 2520, 2520, 1512,  504,  72,  1;
%e A155864   1, 90, 720, 2520, 5040, 6300, 5040, 2520, 720, 90, 1;
%e A155864   ...
%t A155864 (* First program *)
%t A155864 p[n_, x_]:= p[n,x]= If[n==0, 1, 1 + x^n + x*D[(x+1)^(n), {x, 2}]];
%t A155864 Flatten[Table[CoefficientList[p[n,x], x], {n, 0, 12}]]
%t A155864 (* Second program *)
%t A155864 Table[If[k==0 || k==n, 1, 2*Binomial[n, 2]*Binomial[n-2, k-1]], {n,0,12}, {k,0,n}] //Flatten (* _G. C. Greubel_, Jun 04 2021 *)
%o A155864 (Maxima) T(n, k) := ratcoef(x^n + n*(n-1)*x*(x+1)^(n-2) + (1 + (-1)^(2^n))/2, x, k)$ create_list(T(n, k), n, 0, 12, k, 0, n); /* _Franck Maminirina Ramaharo_, Dec 04 2018 */
%o A155864 (Magma)
%o A155864 A155864:= func< n,k | k eq 0 or k eq n select 1 else n*(n-1)*Binomial(n-2, k-1) >;
%o A155864 [A155864(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 04 2021
%o A155864 (Sage)
%o A155864 def A155864(n,k): return 1 if (k==0 or k==n) else n*(n-1)*binomial(n-2,k-1)
%o A155864 flatten([[A155864(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 04 2021
%Y A155864 Cf. A001815, A155863, A155865.
%K A155864 nonn,tabl,easy
%O A155864 0,5
%A A155864 _Roger L. Bagula_, Jan 29 2009
%E A155864 Edited and name clarified by _Franck Maminirina Ramaharo_, Dec 04 2018