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Triangle T(n, k) = coefficients of p(n, x), where p(n, x) = (-1)^n*(1+x)*((n+1)^2 +x)^(n-1), p(0, x) = 1, and p(1, x) = -1-x, read by rows.
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%I #5 May 14 2021 18:47:44

%S 1,-1,-1,9,10,1,-256,-288,-33,-1,15625,17500,1950,76,1,-1679616,

%T -1866240,-194400,-7920,-145,-1,282475249,311299254,30000495,1200500,

%U 24255,246,1,-68719476736,-75161927680,-6694109184,-256901120,-5304320,-61824,-385,-1

%N Triangle T(n, k) = coefficients of p(n, x), where p(n, x) = (-1)^n*(1+x)*((n+1)^2 +x)^(n-1), p(0, x) = 1, and p(1, x) = -1-x, read by rows.

%H G. C. Greubel, <a href="/A158286/b158286.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = coefficients of the characteristic polynomials from the matrix defined by M = M_{0} * M_{1} where M_{0} = (m0_{i,j}), m0_{j,j} = n, otherwise -1 and M_{1} = (m1_{i,j}), m1_{j,j} = -n, otherwise 1.

%F T(n, k) = coefficients of p(n, x), where p(n, x) = (-1)^n*(1+x)*((n+1)^2 +x)^(n-1), p(0, x) = 1, and p(1, x) = -1-x. - _G. C. Greubel_, May 14 2021

%e Triangle begins as:

%e 1;

%e -1, -1;

%e 9, 10, 1;

%e -256, -288, -33, -1;

%e 15625, 17500, 1950, 76, 1;

%e -1679616, -1866240, -194400, -7920, -145, -1;

%e 282475249, 311299254, 30000495, 1200500, 24255, 246, 1;

%e -68719476736, -75161927680, -6694109184, -256901120, -5304320, -61824, -385, -1;

%t (* First program *)

%t M0[n_]:= Table[If[m==k, n, -1], {k,1,n}, {m,1,n}];

%t M1[n_]:= Table[If[m==k, -n, 1], {k,1,n}, {m,1,n}];

%t M[n_]:= M0[n].M1[n];

%t Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n,10}]]//Flatten (* modified by _G. C. Greubel_, May 14 2021 *)

%t (* Second program *)

%t f[n_]:= If[n<2, (-1)^n*(1+n*x), (-1)^n*(1+x)*((n+1)^2 +x)^(n-1)];

%t T[n_, k_]:= SeriesCoefficient[f[n], {x,0,k}];

%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 14 2021 *)

%o (Sage)

%o def p(n,x): return (-1)^n*(1 + n*x) if (n<2) else (-1)^n*(1+x)*((n+1)^2 +x)^(n-1)

%o def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)

%o flatten([T(n) for n in (0..10)]) # _G. C. Greubel_, May 14 2021

%Y Cf. A158285.

%K sign,tabl,less

%O 0,4

%A _Roger L. Bagula_, Mar 15 2009

%E Edited by _G. C. Greubel_, May 14 2021