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

0,5

Links

Table of n, a(n) for n=0..77.

Example

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 ...
    ...

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}]]

Prog

(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 */

Crossrefs

Cf. A173264, A173266.

Sequence in context: A103462 A225641 A116855 * A157744 A335545 A334997

Adjacent sequences: A173262 A173263 A173264 * A173266 A173267 A173268

Keyword

nonn,easy,tabl

Author

Roger L. Bagula, Feb 14 2010

Extensions

Edited by Franck Maminirina Ramaharo, Jan 23 2019

Status

approved