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%I #10 Jan 23 2019 08:29:49
%S 1,1,1,1,2,1,1,2,3,1,1,2,6,4,1,1,2,9,10,5,1,1,2,12,20,15,6,1,1,2,15,
%T 34,35,21,7,1,1,2,18,52,70,56,28,8,1,1,2,21,74,125,126,84,36,9,1,1,2,
%U 24,100,205,252,210,120,45,10,1,1,2,27,130,315,461,462,330,165,55,11,1
%N T(0,k) = 1 and T(n,k) = [x^k] (1 - x^(n + 1))/(1 - 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 3 6 9 12 15 18 21 24 ...
%e 3 | 1 4 10 20 34 52 74 100 130 ...
%e 4 | 1 5 15 35 70 125 205 315 460 ...
%e 5 | 1 6 21 56 126 252 461 786 1266 ...
%e 6 | 1 7 28 84 210 462 924 1715 2996 ...
%e 7 | 1 8 36 120 330 792 1716 3432 6434 ...
%e 8 | 1 9 45 165 495 1287 3003 6435 12870 ...
%e ...
%t p[x_, n_] = If[n == 0, 1/(1 - x), (Sum[x^i, {i, 0, n}])/(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 (1 - x^(n + 1))/(1 - 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 18 2019 */
%Y Cf. A173264, A173266.
%K nonn,easy,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 14 2010
%E Edited by _Franck Maminirina Ramaharo_, Jan 23 2019
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