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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).
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%I #10 Jan 23 2019 08:29:37

%S 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,

%T -2,-3,-14,13,-11,5,-1,1,2,4,22,-20,24,-16,6,-1,1,-2,-5,-32,27,-46,40,

%U -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

%N 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).

%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 1 0 -1 2 -3 4 -5 6 ...

%e 3 | -1 2 -4 8 -14 22 -32 44 -58 ...

%e 4 | -1 3 -7 13 -20 27 -33 37 -38 ...

%e 5 | -1 4 -11 24 -46 82 -139 226 -354 ...

%e 6 | -1 5 -16 40 -86 166 -294 485 -754 ...

%e 7 | -1 6 -22 62 -148 314 -610 1108 -1910 ...

%e 8 | -1 7 -29 91 -239 553 -1163 2269 -4164 ...

%e ...

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

%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 - 2)*x^n + 1)/((x - 1)*(x + 1)^n), 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. A173265, A173266.

%K sign,easy,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 14 2010

%E Edited by _Franck Maminirina Ramaharo_, Jan 23 2019