%I #16 Jan 05 2024 12:49:38
%S 0,-1,0,-16,-27,-120,-81,-128,-485,-1440,-256,-375,-1248,-3607,-8160,
%T -625,-864,-2589,-7264,-16329,-31200,-1296,-1715,-4712,-12843,-28640,
%U -54611,-93240,-2401,-3072,-7845,-20800,-45993,-87456,-149197,-235200,-4096,-5103,-12240,-31615,-69312,-131391,-223888,-352815,-524160
%N Triangle T(n, k) = k^4 - n^4 + 2*k*n*(1 - k^2*n^2), read by rows.
%C A triangular sequence based on the omega(3) Jacobian Elliptic Modular equation.
%H G. C. Greubel, <a href="/A123963/b123963.txt">Rows n = 0..100 of the triangle, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ModularEquation.html">Modular Equation</a>
%F T(n, k) = k^4 - n^4 + 2*k*n*(1 - k^2*n^2).
%F Sum_{k=0..n} T(n, k) = (-1/15)*binomial(n+1, 2) * (15*n^5 +15*n^4 +24*n^3 -9*n^2 -31*n +1). - _G. C. Greubel_, Feb 20 2021
%e Triangular sequence:
%e 0;
%e -1, 0;
%e -16, -27, -120;
%e -81, -128, -485, -1440;
%e -256, -375, -1248, -3607, -8160;
%e -625, -864, -2589, -7264, -16329, -31200;
%e -1296, -1715, -4712, -12843, -28640, -54611, -93240;
%e -2401, -3072, -7845, -20800, -45993, -87456, -149197, -235200;
%e -4096, -5103, -12240, -31615, -69312, -131391, -223888, -352815, -524160;
%t T[n_, k_]:= k^4 - n^4 + 2*n*k*(1 - k^2*n^2);
%t Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by _G. C. Greubel_, Feb 20 2021 *)
%o (Sage) flatten([[k^4 - n^4 + 2*n*k*(1 - k^2*n^2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 20 2021
%o (Magma) [k^4 - n^4 + 2*n*k*(1 - k^2*n^2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 20 2021
%K sign,tabl
%O 0,4
%A _Roger L. Bagula_, Oct 28 2006
%E Edited by _G. C. Greubel_, Feb 20 2021