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%I #13 Feb 20 2022 02:12:13
%S 1,1,1,3,2,1,20,10,3,1,210,84,21,4,1,3003,1001,220,36,5,1,54264,15504,
%T 3060,455,55,6,1,1184040,296010,53130,7315,816,78,7,1,30260340,
%U 6724520,1107568,142506,14950,1330,105,8,1,886163135,177232627,26978328,3262623,324632,27405,2024,136,9,1
%N Triangle, read by rows, where T(n,k) = C(n*(n-1)/2 - k*(k-1)/2 + n-k, n-k).
%C Remarkably, the following matrix products are all equal to A107876: A107862^-1*A107867 = A107867^-1*A107870 = A107870^-1*A107873.
%H G. C. Greubel, <a href="/A107862/b107862.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n,k) = binomial( (n-k)*(n+k+1)/2, n-k). - _G. C. Greubel_, Feb 19 2022
%e Triangle begins:
%e 1;
%e 1, 1;
%e 3, 2, 1;
%e 20, 10, 3, 1;
%e 210, 84, 21, 4, 1;
%e 3003, 1001, 220, 36, 5, 1;
%e 54264, 15504, 3060, 455, 55, 6, 1;
%e 1184040, 296010, 53130, 7315, 816, 78, 7, 1; ...
%t T[n_,k_]:= Binomial[(n-k)*(n+k+1)/2, n-k];
%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 19 2022 *)
%o (PARI) T(n,k)=binomial(n*(n-1)/2-k*(k-1)/2+n-k,n-k)
%o (Magma) [Binomial(Floor((n-k)*(n+k+1)/2), n-k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 19 2022
%o (Sage) flatten([[binomial( (n-k)*(n+k+1)/2, n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 19 2022
%Y Cf. A014068 (column 0), A107863 (column 1), A099121 (column 2), A107865, A107867, A107870, A107876.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Jun 04 2005
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