A123963 - OEIS

%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