%I #8 Mar 27 2021 15:30:53
%S -1,-1,-1,-1,1,-1,-1,5,5,-1,-1,11,29,11,-1,-1,19,89,89,19,-1,-1,29,
%T 209,379,209,29,-1,-1,41,419,1189,1189,419,41,-1,-1,55,755,3079,4829,
%U 3079,755,55,-1,-1,71,1259,6971,15749,15749,6971,1259,71,-1,-1,89,1979,14279,43889,63251,43889,14279,1979,89,-1
%N Triangle T(n, k) = binomial(n, k)^2 - binomial(n, k) - 1, read by rows.
%H G. C. Greubel, <a href="/A144403/b144403.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = binomial(n, k)^2 - binomial(n, k) - 1.
%F Sum_{k=0..n} T(n,k) = Binomial(2*n, n) - 2^n - n - 1. - _G. C. Greubel_, Mar 27 2021
%e Triangle begins as:
%e -1;
%e -1, -1;
%e -1, 1, -1;
%e -1, 5, 5, -1;
%e -1, 11, 29, 11, -1;
%e -1, 19, 89, 89, 19, -1;
%e -1, 29, 209, 379, 209, 29, -1;
%e -1, 41, 419, 1189, 1189, 419, 41, -1;
%e -1, 55, 755, 3079, 4829, 3079, 755, 55, -1;
%e -1, 71, 1259, 6971, 15749, 15749, 6971, 1259, 71, -1;
%e -1, 89, 1979, 14279, 43889, 63251, 43889, 14279, 1979, 89, -1;
%p A144403:= (n,k)-> binomial(n, k)^2 - binomial(n, k) - 1;
%p seq(seq(A144403(n,k), k=0..n), n=0..12); # _G. C. Greubel_, Mar 27 2021
%t Table[Binomial[n, m]^2 -Binomial[n, m] -1, {n,0,12}, {m,0,n}]//Flatten
%o (Magma) [Binomial(n, k)^2 - Binomial(n, k) - 1: k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 27 2021
%o (Sage) flatten([[binomial(n, k)^2 - binomial(n, k) - 1 for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 27 2021
%Y Cf. A000984.
%K sign,tabl
%O 0,8
%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 03 2008
%E Edited by _G. C. Greubel_, Mar 27 2021