A176293 Triangle T(n,k) = 1 + 2*k*(n-k)*(n-1)^2, read by rows.

1, 1, 1, 1, 3, 1, 1, 17, 17, 1, 1, 55, 73, 55, 1, 1, 129, 193, 193, 129, 1, 1, 251, 401, 451, 401, 251, 1, 1, 433, 721, 865, 865, 721, 433, 1, 1, 687, 1177, 1471, 1569, 1471, 1177, 687, 1, 1, 1025, 1793, 2305, 2561, 2561, 2305, 1793, 1025, 1, 1, 1459, 2593, 3403, 3889, 4051, 3889, 3403, 2593, 1459, 1

Offset

0,5

Comments

Row sums are {1, 2, 5, 36, 185, 646, 1757, 4040, 8241, 15370, 26741, ...} = (n+1)*(n^4 - 3*n^3 + 3*n^2 - n + 3)/3.

Links

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

Formula

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

Example

Triangle begins as:
  1;
  1,    1;
  1,    3,    1;
  1,   17,   17,    1;
  1,   55,   73,   55,    1;
  1,  129,  193,  193,  129,    1;
  1,  251,  401,  451,  401,  251,    1;
  1,  433,  721,  865,  865,  721,  433,    1;
  1,  687, 1177, 1471, 1569, 1471, 1177,  687,    1;
  1, 1025, 1793, 2305, 2561, 2561, 2305, 1793, 1025,    1;
  1, 1459, 2593, 3403, 3889, 4051, 3889, 3403, 2593, 1459, 1;

Maple

seq(seq(1 + 2*k*(n-k)*(n-1)^2, k=0..n), n=0..12); # G. C. Greubel, Nov 25 2019

Mathematica

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

Prog

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

Crossrefs

Sequence in context: A290311 A322790 A333560 * A176339 A121412 A212855

Adjacent sequences: A176290 A176291 A176292 * A176294 A176295 A176296

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Apr 14 2010

Extensions

Edited by R. J. Mathar, May 04 2013

Status

approved