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).
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, 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, 24, 100, 205, 252, 210, 120, 45, 10, 1, 1, 2, 27, 130, 315, 461, 462, 330, 165, 55, 11, 1
Offset: 0
Examples
Square array begins:
n\k | 0 1 2 3 4 5 6 7 8 ...
----------------------------------------------
0 | 1 1 1 1 1 1 1 1 1 ...
1 | 1 2 2 2 2 2 2 2 2 ...
2 | 1 3 6 9 12 15 18 21 24 ...
3 | 1 4 10 20 34 52 74 100 130 ...
4 | 1 5 15 35 70 125 205 315 460 ...
5 | 1 6 21 56 126 252 461 786 1266 ...
6 | 1 7 28 84 210 462 924 1715 2996 ...
7 | 1 8 36 120 330 792 1716 3432 6434 ...
8 | 1 9 45 165 495 1287 3003 6435 12870 ...
...
Programs
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Mathematica
p[x_, n_] = If[n == 0, 1/(1 - x), (Sum[x^i, {i, 0, n}])/(1 - x)^n]; a = Table[Table[SeriesCoefficient[Series[p[x, n], {x, 0, 50}], m], {m, 0, 20}], {n, 0, 20}]; Flatten[Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}]] -
Maxima
(kk : 50, nn : 15)$ gf(n) := taylor(if n = 0 then 1/(1 - x) else (1 - x^(n + 1))/(1 - x)^(n + 1), x, 0, kk)$ T(n, k) := ratcoef(gf(n), x, k)$ create_list(T(k, n - k), n, 0, nn, k, 0, n); /* Franck Maminirina Ramaharo, Jan 18 2019 */
Extensions
Edited by Franck Maminirina Ramaharo, Jan 23 2019