A176286 Triangle T(n,k) = 1 + 2*k*(n-k)*(k^2 -n*k +2*n^2) read by rows.

1, 1, 1, 1, 15, 1, 1, 65, 65, 1, 1, 175, 225, 175, 1, 1, 369, 529, 529, 369, 1, 1, 671, 1025, 1135, 1025, 671, 1, 1, 1105, 1761, 2065, 2065, 1761, 1105, 1, 1, 1695, 2785, 3391, 3585, 3391, 2785, 1695, 1, 1, 2465, 4145, 5185, 5681, 5681, 5185, 4145, 2465, 1

Offset

0,5

Comments

This could be written T(n,k) = 1-(n-k)^4 -k^4 +n^4, the quartic analog of A176284.

Row sums are {1, 2, 17, 132, 577, 1798, 4529, 9864, 19329, 34954, 59345, ...} = (n+1)*(9*n^4 -9*n^3 -n^2 +n +15)/15.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(n,k) = T(n,n-k).

Example

Triangle begins as:
  1;
  1,    1;
  1,   15,    1;
  1,   65,   65,    1;
  1,  175,  225,  175,    1;
  1,  369,  529,  529,  369,    1;
  1,  671, 1025, 1135, 1025,  671,    1;
  1, 1105, 1761, 2065, 2065, 1761, 1105,    1;
  1, 1695, 2785, 3391, 3585, 3391, 2785, 1695,    1;
  1, 2465, 4145, 5185, 5681, 5681, 5185, 4145, 2465,    1;
  1, 3439, 5889, 7519, 8449, 8751, 8449, 7519, 5889, 3439, 1;

Maple

seq(seq(n^4 -(n-k)^4 -k^4 +1, k=0..n), n=0..12); # G. C. Greubel, Nov 25 2019

Mathematica

(* First program *)
f[n_, m_, q_]:= f[n, m, q] = 1 -(n-m)^q -m^q +n^q;
Table[Flatten[Table[Table[f[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 1, 10}]
(* Second program *)
Table[n^4 -(n-k)^4 -k^4 +1, {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 25 2019 *)

Prog

(PARI) T(n, k) = n^4 -(n-k)^4 -k^4 +1; \\ G. C. Greubel, Nov 25 2019
(Magma) [n^4 -(n-k)^4 -k^4 +1: k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 25 2019
(Sage) [[n^4 -(n-k)^4 -k^4 +1 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 25 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> n^4 -(n-k)^4 -k^4 +1 ))); # G. C. Greubel, Nov 25 2019

Crossrefs

Sequence in context: A040225 A070644 A174389 * A111805 A238754 A176226

Adjacent sequences: A176283 A176284 A176285 * A176287 A176288 A176289

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Apr 14 2010

Extensions

Edited by R. J. Mathar, May 03 2013

Status

approved