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A107862 Triangle, read by rows, where T(n,k) = C(n*(n-1)/2 - k*(k-1)/2 + n-k, n-k).

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