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
KEYWORD
AUTHOR
Roger L. Bagula, Apr 20 2010
STATUS
approved