A165447 T(n,k) = n^4 - 2*k^2*n^2 + k^4 = A120070(n, k)^2.

9, 64, 25, 225, 144, 49, 576, 441, 256, 81, 1225, 1024, 729, 400, 121, 2304, 2025, 1600, 1089, 576, 169, 3969, 3600, 3025, 2304, 1521, 784, 225, 6400, 5929, 5184, 4225, 3136, 2025, 1024, 289, 9801, 9216, 8281, 7056, 5625, 4096, 2601, 1296, 361, 14400, 13689, 12544, 11025, 9216, 7225, 5184, 3249, 1600, 441, 20449, 19600, 18225, 16384, 14161, 11664, 9025, 6400, 3969, 1936, 529

Offset

2,1

Links

G. C. Greubel, Rows n=2..100 of triangle, flattened

Formula

a(n) = (R*(R+3)-2*(n-3))^2*(R*(1-R)+2*(n+1))^2/16 where R = floor((sqrt(8*n-15)-1)/2). - Luce ETIENNE, Jun 04 2017

G.f.: (x*(1 + 11*x + 11*x^2 + x^3)*(-1 + y)^4 - 2*(-1 + x)^2*x*(1 + x)*(-1 + y)^2*y*(1 + y) + (-1 + x)^4*y*(1 + 11*y + 11*y^2 + y^3))/((-1 + x)^5*(-1 + y)^5). - Stefano Spezia, Oct 21 2018

Example

Triangle begins:
     9;
    64,   25;
   225,  144,   49;
   576,  441,  256,   81;
  1225, 1024,  729,  400, 121;
  2304, 2025, 1600, 1089, 576, 169;
  ...

Maple

a:=(n, k)->(n^2-k^2)^2: seq(seq(a(n, k), k=1..n-1), n=2..12); # Muniru A Asiru, Oct 21 2018

Mathematica

Table[n^2 - k^2, {n, 2, 12}, {k, n - 1}]^2 // Flatten (* Michael De Vlieger, Jun 06 2017, after Alonso del Arte at A120070 *)
f[n_] := Table[SeriesCoefficient[(x (1 + 11 x + 11 x^2 + x^3) (-1 + y)^4 - 2 (-1 + x)^2 x (1 + x) (-1 + y)^2 y (1 + y) + (-1 + x)^4 y (1 + 11 y + 11 y^2 + y^3))/((-1 + x)^5 (-1 + y)^5) , {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 1, n-1}]; Flatten[Array[f, 11]] (* Stefano Spezia, Oct 21 2018 *)

Prog

(PARI) for (n=2, 10, for(k=1, n-1, print1((n^2-k^2)^2, ", ")); print()); \\ Michel Marcus, Jun 04 2017
(GAP) Flat(List([2..12], n->List([1..n-1], k->n^4-2*k^2*n^2+k^4))); # Muniru A Asiru, Oct 21 2018
(Magma) [(Floor((Sqrt(8*n-15)-1)/2)*(Floor((Sqrt(8*n-15)-1)/2)+3)-2*(n-3))^2*(Floor((Sqrt(8*n-15)-1)/2)*(1-Floor((Sqrt(8*n-15)-1)/2))+2*(n+1))^2/16: n in [2..30]]; // G. C. Greubel, Oct 20 2018

Crossrefs

Cf. A120070.

Sequence in context: A251211 A342197 A302975 * A050792 A171671 A016886

Adjacent sequences: A165444 A165445 A165446 * A165448 A165449 A165450

Keyword

nonn,tabl,easy

Author

Paul Curtz, Sep 20 2009

Extensions

More terms from Michel Marcus, Jun 04 2017

Status

approved