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A176286 Triangle T(n,k) = 1 + 2*k*(n-k)*(k^2 -n*k +2*n^2) read by rows.

%I A176286 #11 Sep 08 2022 08:45:52
%S A176286 1,1,1,1,15,1,1,65,65,1,1,175,225,175,1,1,369,529,529,369,1,1,671,
%T A176286 1025,1135,1025,671,1,1,1105,1761,2065,2065,1761,1105,1,1,1695,2785,
%U A176286 3391,3585,3391,2785,1695,1,1,2465,4145,5185,5681,5681,5185,4145,2465,1
%N A176286 Triangle T(n,k) = 1 + 2*k*(n-k)*(k^2 -n*k +2*n^2) read by rows.
%C A176286 This could be written T(n,k) = 1-(n-k)^4 -k^4 +n^4, the quartic analog of A176284.
%C A176286 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.
%H A176286 G. C. Greubel, <a href="/A176286/b176286.txt">Rows n = 0..100 of triangle, flattened</a>
%F A176286 T(n,k) = T(n,n-k).
%e A176286 Triangle begins as:
%e A176286   1;
%e A176286   1,    1;
%e A176286   1,   15,    1;
%e A176286   1,   65,   65,    1;
%e A176286   1,  175,  225,  175,    1;
%e A176286   1,  369,  529,  529,  369,    1;
%e A176286   1,  671, 1025, 1135, 1025,  671,    1;
%e A176286   1, 1105, 1761, 2065, 2065, 1761, 1105,    1;
%e A176286   1, 1695, 2785, 3391, 3585, 3391, 2785, 1695,    1;
%e A176286   1, 2465, 4145, 5185, 5681, 5681, 5185, 4145, 2465,    1;
%e A176286   1, 3439, 5889, 7519, 8449, 8751, 8449, 7519, 5889, 3439, 1;
%p A176286 seq(seq(n^4 -(n-k)^4 -k^4 +1, k=0..n), n=0..12); # _G. C. Greubel_, Nov 25 2019
%t A176286 (* First program *)
%t A176286 f[n_, m_, q_]:= f[n, m, q] = 1 -(n-m)^q -m^q +n^q;
%t A176286 Table[Flatten[Table[Table[f[n, m, q], {m, 0, n}], {n, 0, 10}]], {q,1,10}]
%t A176286 (* Second program *)
%t A176286 Table[n^4 -(n-k)^4 -k^4 +1, {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 25 2019 *)
%o A176286 (PARI) T(n,k) = n^4 -(n-k)^4 -k^4 +1; \\ _G. C. Greubel_, Nov 25 2019
%o A176286 (Magma) [n^4 -(n-k)^4 -k^4 +1: k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 25 2019
%o A176286 (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
%o A176286 (GAP) Flat(List([0..12], n-> List([0..n], k-> n^4 -(n-k)^4 -k^4 +1 ))); # _G. C. Greubel_, Nov 25 2019
%K A176286 nonn,tabl,easy
%O A176286 0,5
%A A176286 _Roger L. Bagula_, Apr 14 2010
%E A176286 Edited by _R. J. Mathar_, May 03 2013