A176566 Triangle T(n, k) = binomial(n*(n+1)/2 + k, k), read by rows.

1, 1, 1, 1, 2, 3, 1, 4, 10, 20, 1, 7, 28, 84, 210, 1, 11, 66, 286, 1001, 3003, 1, 16, 136, 816, 3876, 15504, 54264, 1, 22, 253, 2024, 12650, 65780, 296010, 1184040, 1, 29, 435, 4495, 35960, 237336, 1344904, 6724520, 30260340, 1, 37, 703, 9139, 91390, 749398, 5245786, 32224114, 177232627, 886163135

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flatten

Formula

T(n, k) = binomial(binomial(n, 2) + k, k).

Sum_{k=0..n} T(n, k) = A107868(n).

Example

Square array of T(n, k):
  1,  1,   1,    1,     1,     1,      1 ...
  1,  1,   1,    1,     1,     1,      1 ... A000012;
  1,  2,   3,    4,     5,     6,      7 ... A000027;
  1,  4,  10,   20,    35,    56,     84 ... A000292;
  1,  7,  28,   84,   210,   462,    924 ... A000579;
  1, 11,  66,  286,  1001,  3003,   8008 ... A001287;
  1, 16, 136,  816,  3876, 15504,  54264 ... A010968;
  1, 22, 253, 2024, 12650, 65780, 296010 ... A010974;
Triangle begins as:
  1;
  1,  1;
  1,  2,   3;
  1,  4,  10,   20;
  1,  7,  28,   84,   210;
  1, 11,  66,  286,  1001,   3003;
  1, 16, 136,  816,  3876,  15504,   54264;
  1, 22, 253, 2024, 12650,  65780,  296010,  1184040;
  1, 29, 435, 4495, 35960, 237336, 1344904,  6724520,  30260340;
  1, 37, 703, 9139, 91390, 749398, 5245786, 32224114, 177232627, 886163135;

Mathematica

T[n_, k_]= Binomial[Binomial[n, 2] + k, k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Magma) [Binomial(Binomial(n, 2) + k, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2021
(Sage) flatten([[binomial(binomial(n, 2) +k, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 09 2021
(PARI) row(n) = vector(n+1, k, k--; binomial(binomial(n, 2) + k, k)); \\ Michel Marcus, Jul 10 2021

Crossrefs

Cf. A107868 (rows sums), A158498.

Sequence in context: A211233 A084608 A078990 * A079639 A104694 A125182

Adjacent sequences: A176563 A176564 A176565 * A176567 A176568 A176569

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Apr 20 2010

Status

approved