%I #9 Jan 23 2019 08:29:23

%S 1,1,-1,1,-2,-1,1,-2,0,-1,1,-2,-2,0,-1,1,-2,2,0,0,-1,1,-2,-4,-2,0,0,

%T -1,1,-2,6,2,0,0,0,-1,1,-2,-10,0,-2,0,0,0,-1,1,-2,16,-4,2,0,0,0,0,-1,

%U 1,-2,-26,6,0,-2,0,0,0,0,-1,1,-2,42,-2,0,2,0,0,0,0,0,-1

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

%e Square array begins:

%e n\k | 0 1 2 3 4 5 6 7 8 ...

%e --------------------------------------------

%e 0 | 1 1 1 1 1 1 1 1 1 ...

%e 1 | -1 -2 -2 -2 -2 -2 -2 -2 -2 ...

%e 2 | -1 0 -2 2 -4 6 -10 16 -26 ...

%e 3 | -1 0 0 -2 2 0 -4 6 -2 ...

%e 4 | -1 0 0 0 -2 2 0 0 -4 ...

%e 5 | -1 0 0 0 0 -2 2 0 0 ...

%e 6 | -1 0 0 0 0 0 -2 2 0 ...

%e 7 | -1 0 0 0 0 0 0 -2 2 ...

%e 8 | -1 0 0 0 0 0 0 0 -2 ...

%e ...

%t p[x_, n_] = If[n == 0, 1/(1 - x), (Sum[x^i, {i, 0, n}])/(x^n - Sum[x^i, {i, 0, n - 1}])];

%t a = Table[Table[SeriesCoefficient[Series[p[x, n], {x, 0, 50}], m], {m, 0, 20}], {n, 0, 20}];

%t Flatten[Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}]]

%o (Maxima) (kk : 50, nn : 15)$

%o gf(n) := taylor(if n = 0 then 1/(1 - x) else (x^(n + 1) - 1)/((x - 2)*x^n + 1), x, 0, kk)$

%o T(n, k) := ratcoef(gf(n), x, k)$

%o create_list(T(k, n - k), n, 0, nn, k, 0, n);

%o /* _Franck Maminirina Ramaharo_, Jan 23 2019 */

%Y Cf. A173264, A173265.

%K sign,easy,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 14 2010

%E Edited by _Franck Maminirina Ramaharo_, Jan 23 2019