A173264 T(0,k) = 1 and T(n,k) = [x^k] ((x - 2)*x^n + 1)/((x - 1)*(x + 1)^n) for n >= 1, square array read by descending antidiagonals (n >= 0, k >= 0).

1, 1, -1, 1, 2, -1, 1, -2, 1, -1, 1, 2, 0, 2, -1, 1, -2, -1, -4, 3, -1, 1, 2, 2, 8, -7, 4, -1, 1, -2, -3, -14, 13, -11, 5, -1, 1, 2, 4, 22, -20, 24, -16, 6, -1, 1, -2, -5, -32, 27, -46, 40, -22, 7, -1, 1, 2, 6, 44, -33, 82, -86, 62, -29, 8, -1, 1, -2, -7, -58, 37, -139, 166, -148, 91, -37, 9, -1

Offset: 0

Author

Roger L. Bagula, Feb 14 2010

Keywords

• sign
• easy
• tabl

Examples

			Square array begins:
  n\k |  0   1   2   3    4   5     6    7     8 ...
  --------------------------------------------------
    0 |  1   1   1   1    1   1     1    1     1 ...
    1 | -1   2  -2   2   -2   2    -2    2    -2 ...
    2 | -1   1   0  -1    2  -3     4   -5     6 ...
    3 | -1   2  -4   8  -14  22   -32   44   -58 ...
    4 | -1   3  -7  13  -20  27   -33   37   -38 ...
    5 | -1   4 -11  24  -46  82  -139  226  -354 ...
    6 | -1   5 -16  40  -86 166  -294  485  -754 ...
    7 | -1   6 -22  62 -148 314  -610 1108 -1910 ...
    8 | -1   7 -29  91 -239 553 -1163 2269 -4164 ...
    ...
		

Crossrefs

Cf. A173265, A173266.

Programs

Mathematica

p[x_, n_] = If[n == 0, 1/(1 - x), (x^n - Sum[x^i, {i, 0, n - 1}])/( 1 + x)^n];
a = Table[Table[SeriesCoefficient[Series[p[x, n], {x, 0, 50}], m], {m, 0, 20}], {n, 0, 20}];
Flatten[Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}]]

Maxima

(kk : 50, nn : 15)$
gf(n) := taylor(if n = 0 then 1/(1 - x) else ((x - 2)*x^n + 1)/((x - 1)*(x + 1)^n), x, 0, kk)$
T(n, k) := ratcoef(gf(n), x, k)$
create_list(T(k, n - k), n, 0, nn, k, 0, n);
/* Franck Maminirina Ramaharo, Jan 23 2019 */

Extensions

Edited by Franck Maminirina Ramaharo, Jan 23 2019