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A176286 Triangle T(n,k) = 1 + 2*k*(n-k)*(k^2 -n*k +2*n^2) read by rows. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified August 10 08:35 EDT 2020. Contains 336368 sequences. (Running on oeis4.)