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A176283 Triangle T(n,k) = 1 + A000537(n) - A000537(k) - A000537(n-k), read by rows. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified July 4 18:38 EDT 2020. Contains 335448 sequences. (Running on oeis4.)