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

Author

Roger L. Bagula, Apr 20 2010

Keywords

• nonn
• tabl
• easy

Examples

			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;
		

Links

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

Crossrefs

Cf. A107868 (rows sums), A158498.

Programs

Magma

[Binomial(Binomial(n, 2) + k, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2021

Mathematica

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

PARI

row(n) = vector(n+1, k, k--; binomial(binomial(n,2) + k, k)); \\ Michel Marcus, Jul 10 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

Formula

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

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