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

%I #12 Jan 22 2019 08:47:13

%S -1,-1,-1,-1,-1,-1,-1,-1,0,-1,-1,-1,-1,1,-1,-1,-1,-1,0,1,-1,-1,-1,-1,

%T -1,1,0,-1,-1,-1,-1,-1,0,2,-1,-1,-1,-1,-1,-1,-1,1,2,-1,-1,-1,-1,-1,-1,

%U -1,0,2,1,0,-1,-1,-1,-1,-1,-1,-1,1,3,-1,1,-1,-1,-1,-1,-1,-1,-1,0,2,3,-3,1,-1

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

%e Array begins:

%e n\k | 0 1 2 3 4 5 6 7 8 9 10 ...

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

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

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

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

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

%e 5 | -1 -1 -1 -1 -1 0 1 2 3 4 4 ...

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

%e 7 | -1 -1 -1 -1 -1 -1 -1 0 1 2 3 ...

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

%e 9 | -1 -1 -1 -1 -1 -1 -1 -1 -1 0 1 ...

%e 10 | -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 ...

%e ...

%t f[t_, n_] = If[n == 1, 1/(-1 + t), 1/(-1 + t - t^n)];

%t a = Table[Table[SeriesCoefficient[Series[f[t, m], {t, 0, 30}], n], {n, 0, 30}], {m, 1, 31}];

%t Flatten[Table[Table[a[[n - m + 1]][[m]], {m, 1, n }], {n, 1, 15}]]

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

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

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

%o create_list(T(n - k, k), n, 1, nn, k, 0, n - 1);

%o /* _Franck Maminirina Ramaharo_, Jan 21 2019 */

%Y Cf. A144383, A144384.

%K sign,easy,tabl

%O 1,34

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 01 2008

%E Edited by _Franck Maminirina Ramaharo_, Jan 21 2019