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A176293 Triangle T(n,k) = 1 + 2*k*(n-k)*(n-1)^2, read by rows. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified October 15 21:38 EDT 2021. Contains 348034 sequences. (Running on oeis4.)