|
%I #14 Feb 24 2021 08:14:55
%S 0,-4,0,-8,1,28,-12,4,52,132,-16,9,84,209,384,-20,16,124,304,556,880,
%T -24,25,172,417,760,1201,1740,-28,36,228,548,996,1572,2276,3108,-32,
%U 49,292,697,1264,1993,2884,3937,5152,-36,64,364,864,1564,2464,3564,4864,6364,8064,-40,81,444,1049,1896,2985,4316,5889,7704,9761,12060
%N Triangle T(n, k) = k^2*(1+n)^2 - 4*n, read by rows.
%C A triangular sequence formed from the omega2 Jacobian Elliptic Modular function.
%H G. C. Greubel, <a href="/A123961/b123961.txt">Table of n, a(n) for n = 0..5150</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ModularEquation.html">Modular Equation</a>
%F T(n, k) = k^2*(1+n)^2 - 4*n.
%F Sum_{k=0..n} T(n, k) = (n*(n+1)/6)*( 2*n^3 + 5*n^2 + 4*n - 23 ). (n+1)^2 * A000330(n) - 8 * A000217(n). - _G. C. Greubel_, Feb 19 2021
%e Triangle begins:
%e 0;
%e -4, 0;
%e -8, 1, 28;
%e -12, 4, 52, 132;
%e -16, 9, 84, 209, 384;
%e -20, 16, 124, 304, 556, 880;
%e -24, 25, 172, 417, 760, 1201, 1740;
%e -28, 36, 228, 548, 996, 1572, 2276, 3108;
%e -32, 49, 292, 697, 1264, 1993, 2884, 3937, 5152;
%e -36, 64, 364, 864, 1564, 2464, 3564, 4864, 6364, 8064;
%e -40, 81, 444, 1049, 1896, 2985, 4316, 5889, 7704, 9761, 12060;
%t T[n_, k_]:= k^2*(1+n)^2 - 4*n;
%t Table[T[n, k], {n, 0, 12}, {k,0,n}]//Flatten
%o (Sage) flatten([[k^2*(n+1)^2 - 4*n for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 19 2021
%o (Magma) [k^2*(n+1)^2 - 4*n: k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 19 2021
%Y Cf. A000217, A000330.
%K tabl,sign
%O 0,2
%A _Roger L. Bagula_, Oct 28 2006
%E Edited by _G. C. Greubel_, Feb 19 2021
|
