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%I #12 Sep 08 2022 08:45:52
%S 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,
%T 333,216,1,1,343,551,649,649,551,343,1,1,512,847,1036,1097,1036,847,
%U 512,1,1,729,1233,1549,1701,1701,1549,1233,729,1,1,1000,1721,2206,2485,2576,2485,2206,1721,1000,1
%N Triangle T(n,k) = 1 + A000537(n) - A000537(k) - A000537(n-k), read by rows.
%C Like A176282 but build on sums of cubes (A000537) instead of sums of squares.
%C 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.
%H G. C. Greubel, <a href="/A176283/b176283.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n,k) = T(n,n-k).
%F 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
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 8, 1;
%e 1, 27, 27, 1;
%e 1, 64, 83, 64, 1;
%e 1, 125, 181, 181, 125, 1;
%e 1, 216, 333, 370, 333, 216, 1;
%e 1, 343, 551, 649, 649, 551, 343, 1;
%e 1, 512, 847, 1036, 1097, 1036, 847, 512, 1;
%e 1, 729, 1233, 1549, 1701, 1701, 1549, 1233, 729, 1;
%e 1, 1000, 1721, 2206, 2485, 2576, 2485, 2206, 1721, 1000, 1;
%p seq(seq(, k=0..n), n=0..12); # _G. C. Greubel_, Nov 25 2019
%t (* sequences with q=1..10 *)
%t 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}]
%t (* Second program *)
%t 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 *)
%o (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
%o (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
%o (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
%o (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
%K nonn,tabl,easy
%O 0,5
%A _Roger L. Bagula_, Apr 14 2010
%E Edited by _R. J. Mathar_, May 03 2013
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