A158285 Triangle T(n, k) = coefficients of p(n, x), where p(n, x) = (-1)^n*(x+2-n)*(x+2)^(n-1), p(0, x) = 1, and p(1, x) = -1-x, read by rows.

1, -1, -1, 0, 2, 1, 4, 0, -3, -1, -16, -16, 0, 4, 1, 48, 80, 40, 0, -5, -1, -128, -288, -240, -80, 0, 6, 1, 320, 896, 1008, 560, 140, 0, -7, -1, -768, -2560, -3584, -2688, -1120, -224, 0, 8, 1, 1792, 6912, 11520, 10752, 6048, 2016, 336, 0, -9, -1, -4096, -17920, -34560, -38400, -26880, -12096, -3360, -480, 0, 10, 1

Offset

0,5

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_{i,j}), m_{j,j} = -1, else 1.

T(n, k) = coefficients of p(n, x), where p(n, x) = (-1)^n*(x+2-n)*(x+2)^(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;
      0,      2,      1;
      4,      0,     -3,     -1;
    -16,    -16,      0,      4,      1;
     48,     80,     40,      0,     -5,     -1;
   -128,   -288,   -240,    -80,      0,      6,     1;
    320,    896,   1008,    560,    140,      0,    -7,   -1;
   -768,  -2560,  -3584,  -2688,  -1120,   -224,     0,    8,  1;
   1792,   6912,  11520,  10752,   6048,   2016,   336,    0, -9, -1;
  -4096, -17920, -34560, -38400, -26880, -12096, -3360, -480,  0, 10, 1;

Mathematica

(* First program *)
M[n_]:= Table[If[k==m, -1, 1], {k, 0, n}, {m, 0, n}];
Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 0, 10}]]//Flatten (* modified by G. C. Greubel, May 14 2021 *)
(* Second program *)
f[n_]:= If[n<2, (-1)^n*(1+n*x), (-1)^n*(x+2-n)*(x+2)^(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*(2-n+x)*(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. A158286.

Sequence in context: A048614 A001442 A226952 * A277994 A334112 A355625

Adjacent sequences: A158282 A158283 A158284 * A158286 A158287 A158288

Keyword

sign,tabl,less

Author

Roger L. Bagula, Mar 15 2009

Extensions

Edited by G. C. Greubel, May 14 2021

Status

approved