A158286 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.

1, -1, -1, 9, 10, 1, -256, -288, -33, -1, 15625, 17500, 1950, 76, 1, -1679616, -1866240, -194400, -7920, -145, -1, 282475249, 311299254, 30000495, 1200500, 24255, 246, 1, -68719476736, -75161927680, -6694109184, -256901120, -5304320, -61824, -385, -1

Offset

0,4

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

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.

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

Example

Triangle begins as:
             1;
            -1,           -1;
             9,           10,           1;
          -256,         -288,         -33,         -1;
         15625,        17500,        1950,         76,        1;
      -1679616,     -1866240,     -194400,      -7920,     -145,     -1;
     282475249,    311299254,    30000495,    1200500,    24255,    246,    1;
  -68719476736, -75161927680, -6694109184, -256901120, -5304320, -61824, -385, -1;

Mathematica

(* First program *)
M0[n_]:= Table[If[m==k, n, -1], {k, 1, n}, {m, 1, n}];
M1[n_]:= Table[If[m==k, -n, 1], {k, 1, n}, {m, 1, n}];
M[n_]:= M0[n].M1[n];
Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 10}]]//Flatten (* modified by G. C. Greubel, May 14 2021 *)
(* Second program *)
f[n_]:= If[n<2, (-1)^n*(1+n*x), (-1)^n*(1+x)*((n+1)^2 +x)^(n-1)];
T[n_, k_]:= SeriesCoefficient[f[n], {x, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, May 14 2021 *)

Prog

(Sage)
def p(n, x): return (-1)^n*(1 + n*x) if (n<2) else (-1)^n*(1+x)*((n+1)^2 +x)^(n-1)
def T(n): return ( p(n, x) ).full_simplify().coefficients(sparse=False)
flatten([T(n) for n in (0..10)]) # G. C. Greubel, May 14 2021

Crossrefs

Cf. A158285.

Sequence in context: A109409 A262551 A160563 * A126839 A341818 A379529

Adjacent sequences: A158283 A158284 A158285 * A158287 A158288 A158289

Keyword

sign,tabl,less

Author

Roger L. Bagula, Mar 15 2009

Extensions

Edited by G. C. Greubel, May 14 2021

Status

approved