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%I #14 Sep 08 2022 08:45:52
%S 1,1,1,1,2,3,1,4,10,20,1,7,28,84,210,1,11,66,286,1001,3003,1,16,136,
%T 816,3876,15504,54264,1,22,253,2024,12650,65780,296010,1184040,1,29,
%U 435,4495,35960,237336,1344904,6724520,30260340,1,37,703,9139,91390,749398,5245786,32224114,177232627,886163135
%N Triangle T(n, k) = binomial(n*(n+1)/2 + k, k), read by rows.
%H G. C. Greubel, <a href="/A176566/b176566.txt">Rows n = 0..50 of the triangle, flatten</a>
%F T(n, k) = binomial(binomial(n, 2) + k, k).
%F Sum_{k=0..n} T(n, k) = A107868(n).
%e Square array of T(n, k):
%e 1, 1, 1, 1, 1, 1, 1 ...
%e 1, 1, 1, 1, 1, 1, 1 ... A000012;
%e 1, 2, 3, 4, 5, 6, 7 ... A000027;
%e 1, 4, 10, 20, 35, 56, 84 ... A000292;
%e 1, 7, 28, 84, 210, 462, 924 ... A000579;
%e 1, 11, 66, 286, 1001, 3003, 8008 ... A001287;
%e 1, 16, 136, 816, 3876, 15504, 54264 ... A010968;
%e 1, 22, 253, 2024, 12650, 65780, 296010 ... A010974;
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 2, 3;
%e 1, 4, 10, 20;
%e 1, 7, 28, 84, 210;
%e 1, 11, 66, 286, 1001, 3003;
%e 1, 16, 136, 816, 3876, 15504, 54264;
%e 1, 22, 253, 2024, 12650, 65780, 296010, 1184040;
%e 1, 29, 435, 4495, 35960, 237336, 1344904, 6724520, 30260340;
%e 1, 37, 703, 9139, 91390, 749398, 5245786, 32224114, 177232627, 886163135;
%t T[n_, k_]= Binomial[Binomial[n, 2] + k, k];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
%o (Magma) [Binomial(Binomial(n, 2) + k, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 09 2021
%o (Sage) flatten([[binomial(binomial(n,2) +k, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 09 2021
%o (PARI) row(n) = vector(n+1, k, k--; binomial(binomial(n,2) + k, k)); \\ _Michel Marcus_, Jul 10 2021
%Y Cf. A107868 (rows sums), A158498.
%K nonn,tabl,easy
%O 0,5
%A _Roger L. Bagula_, Apr 20 2010