A123961 Triangle T(n, k) = k^2*(1+n)^2 - 4*n, read by rows.

0, -4, 0, -8, 1, 28, -12, 4, 52, 132, -16, 9, 84, 209, 384, -20, 16, 124, 304, 556, 880, -24, 25, 172, 417, 760, 1201, 1740, -28, 36, 228, 548, 996, 1572, 2276, 3108, -32, 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

Offset

0,2

Comments

A triangular sequence formed from the omega2 Jacobian Elliptic Modular function.

Links

G. C. Greubel, Table of n, a(n) for n = 0..5150

Eric Weisstein's World of Mathematics, Modular Equation

Formula

T(n, k) = k^2*(1+n)^2 - 4*n.

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

Example

Triangle begins:
    0;
   -4,  0;
   -8,  1,  28;
  -12,  4,  52,  132;
  -16,  9,  84,  209,  384;
  -20, 16, 124,  304,  556,  880;
  -24, 25, 172,  417,  760, 1201, 1740;
  -28, 36, 228,  548,  996, 1572, 2276, 3108;
  -32, 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;

Mathematica

T[n_, k_]:= k^2*(1+n)^2 - 4*n;
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(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
(Magma) [k^2*(n+1)^2 - 4*n: k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 19 2021

Crossrefs

Cf. A000217, A000330.

Sequence in context: A288096 A021249 A010638 * A020763 A229911 A244336

Adjacent sequences: A123958 A123959 A123960 * A123962 A123963 A123964

Keyword

tabl,sign

Author

Roger L. Bagula, Oct 28 2006

Extensions

Edited by G. C. Greubel, Feb 19 2021

Status

approved