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%I #15 Sep 08 2022 08:45:52
%S 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,
%T 401,251,1,1,433,721,865,865,721,433,1,1,687,1177,1471,1569,1471,1177,
%U 687,1,1,1025,1793,2305,2561,2561,2305,1793,1025,1,1,1459,2593,3403,3889,4051,3889,3403,2593,1459,1
%N Triangle T(n,k) = 1 + 2*k*(n-k)*(n-1)^2, read by rows.
%C 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.
%H G. C. Greubel, <a href="/A176293/b176293.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n,k) = T(n,n-k) = 1 - (-n^2 - n^4 + (n*k + n - k)^2 + (k + n*(n - k))^2).
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 17, 17, 1;
%e 1, 55, 73, 55, 1;
%e 1, 129, 193, 193, 129, 1;
%e 1, 251, 401, 451, 401, 251, 1;
%e 1, 433, 721, 865, 865, 721, 433, 1;
%e 1, 687, 1177, 1471, 1569, 1471, 1177, 687, 1;
%e 1, 1025, 1793, 2305, 2561, 2561, 2305, 1793, 1025, 1;
%e 1, 1459, 2593, 3403, 3889, 4051, 3889, 3403, 2593, 1459, 1;
%p seq(seq(1 + 2*k*(n-k)*(n-1)^2, k=0..n), n=0..12); # _G. C. Greubel_, Nov 25 2019
%t 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
%o (PARI) T(n,k) = 1 + 2*k*(n-k)*(n-1)^2; \\ _G. C. Greubel_, Nov 25 2019
%o (Magma) [1 + 2*k*(n-k)*(n-1)^2: k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 25 2019
%o (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
%o (GAP) Flat(List([0..12], n-> List([0..n], k-> 1 + 2*k*(n-k)*(n-1)^2 ))); # _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 04 2013
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