A176283 Triangle T(n,k) = 1 + A000537(n) - A000537(k) - A000537(n-k), read by rows.

1, 1, 1, 1, 8, 1, 1, 27, 27, 1, 1, 64, 83, 64, 1, 1, 125, 181, 181, 125, 1, 1, 216, 333, 370, 333, 216, 1, 1, 343, 551, 649, 649, 551, 343, 1, 1, 512, 847, 1036, 1097, 1036, 847, 512, 1, 1, 729, 1233, 1549, 1701, 1701, 1549, 1233, 729, 1, 1, 1000, 1721, 2206, 2485, 2576, 2485, 2206, 1721, 1000, 1

Offset

0,5

Comments

Like A176282 but build on sums of cubes (A000537) instead of sums of squares.

Row sums are {1, 2, 10, 56, 213, 614, 1470, 3088, 5889, 10426, 17402, ...} = (n+1)*(9*n^4 + 6*n^3 - 11*n^2 - 4*n + 60)/60.

Links

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

Formula

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

T(n, k) = (4 + n^2*(n+1)^2 - k^2*(k+1)^2 - (n-k)^2*(n-k+1)^2)/4. - G. C. Greubel, Nov 25 2019

Example

Triangle begins as:
  1;
  1,    1;
  1,    8,    1;
  1,   27,   27,    1;
  1,   64,   83,   64,    1;
  1,  125,  181,  181,  125,    1;
  1,  216,  333,  370,  333,  216,    1;
  1,  343,  551,  649,  649,  551,  343,    1;
  1,  512,  847, 1036, 1097, 1036,  847,  512,    1;
  1,  729, 1233, 1549, 1701, 1701, 1549, 1233,  729,    1;
  1, 1000, 1721, 2206, 2485, 2576, 2485, 2206, 1721, 1000, 1;

Maple

seq(seq(, k=0..n), n=0..12); # G. C. Greubel, Nov 25 2019

Mathematica

(* sequences with q=1..10 *)
f[n_, k_, q_]:= f[n, k, q] = 1 + Sum[i^q, {i, 0, n}] - Sum[i^q, {i, 0, k}] + Sum[i^q, {i, 0, n-k}])); Table[Flatten[Table[f[n, k, q], {n, 0, 10}, {k, 0, n}]], {q, 1, 10}]
(* Second program *)
Table[(4 +n^2*(n+1)^2 -k^2*(k+1)^2 -(n-k)^2*(n-k+1)^2)/4, {n, 0, 12}, {k, 0, n} ]//Flatten (* G. C. Greubel, Nov 25 2019 *)

Prog

(PARI) T(n, k) = 1 + (n^2*(n+1)^2 - k^2*(k+1)^2 - (n-k)^2*(n-k+1)^2)/4; \\ G. C. Greubel, Nov 25 2019
(Magma) [(4 +n^2*(n+1)^2 -k^2*(k+1)^2 -(n-k)^2*(n-k+1)^2)/4: k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 25 2019
(Sage) [[(4 +n^2*(n+1)^2 -k^2*(k+1)^2 -(n-k)^2*(n-k+1)^2)/4 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-> (4 +n^2*(n+1)^2 -k^2*(k+1)^2 - (n-k)^2*(n-k+1)^2)/4  ))); # G. C. Greubel, Nov 25 2019

Crossrefs

Sequence in context: A147295 A174388 A220718 * A323324 A181543 A141696

Adjacent sequences: A176280 A176281 A176282 * A176284 A176285 A176286

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Apr 14 2010

Extensions

Edited by R. J. Mathar, May 03 2013

Status

approved