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A173265 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).

%I A173265 #10 Jan 23 2019 08:29:49
%S A173265 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 A173265 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 A173265 24,100,205,252,210,120,45,10,1,1,2,27,130,315,461,462,330,165,55,11,1
%N A173265 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 A173265 Square array begins:
%e A173265   n\k | 0  1  2   3   4    5    6    7     8 ...
%e A173265   ----------------------------------------------
%e A173265     0 | 1  1  1   1   1    1    1    1     1 ...
%e A173265     1 | 1  2  2   2   2    2    2    2     2 ...
%e A173265     2 | 1  3  6   9  12   15   18   21    24 ...
%e A173265     3 | 1  4 10  20  34   52   74  100   130 ...
%e A173265     4 | 1  5 15  35  70  125  205  315   460 ...
%e A173265     5 | 1  6 21  56 126  252  461  786  1266 ...
%e A173265     6 | 1  7 28  84 210  462  924 1715  2996 ...
%e A173265     7 | 1  8 36 120 330  792 1716 3432  6434 ...
%e A173265     8 | 1  9 45 165 495 1287 3003 6435 12870 ...
%e A173265     ...
%t A173265 p[x_, n_] = If[n == 0, 1/(1 - x), (Sum[x^i, {i, 0, n}])/(1 - x)^n];
%t A173265 a = Table[Table[SeriesCoefficient[Series[p[x, n], {x, 0, 50}], m], {m, 0, 20}], {n, 0, 20}];
%t A173265 Flatten[Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}]]
%o A173265 (Maxima) (kk : 50, nn : 15)$
%o A173265 gf(n) := taylor(if n = 0 then 1/(1 - x) else (1 - x^(n + 1))/(1 - x)^(n + 1), x, 0, kk)$
%o A173265 T(n, k) := ratcoef(gf(n), x, k)$
%o A173265 create_list(T(k, n - k), n, 0, nn, k, 0, n);
%o A173265 /* _Franck Maminirina Ramaharo_, Jan 18 2019 */
%Y A173265 Cf. A173264, A173266.
%K A173265 nonn,easy,tabl
%O A173265 0,5
%A A173265 _Roger L. Bagula_, Feb 14 2010
%E A173265 Edited by _Franck Maminirina Ramaharo_, Jan 23 2019