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%I #6 Sep 08 2022 08:45:52
%S 1,1,1,1,1,1,1,-6,-6,1,1,-119,-154,-119,1,1,-1991,-2651,-2651,-1991,1,
%T 1,-38744,-50252,-52632,-50252,-38744,1,1,-888008,-1118007,-1169366,
%U -1169366,-1118007,-888008,1,1,-23535791,-28915001,-30018509,-30188420,-30018509,-28915001,-23535791,1
%N Triangle T(n, k) = binomial(binomial(n, 2) + k, k) + binomial(binomial(n, 2) + n-k, n-k) - binomial(binomial(n, 2) + n, n), read by rows.
%H G. C. Greubel, <a href="/A176567/b176567.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = binomial(binomial(n, 2) + k, k) + binomial(binomial(n, 2) + n-k, n-k) - binomial(binomial(n, 2) + n, n).
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, -6, -6, 1;
%e 1, -119, -154, -119, 1;
%e 1, -1991, -2651, -2651, -1991, 1;
%e 1, -38744, -50252, -52632, -50252, -38744, 1;
%e 1, -888008, -1118007, -1169366, -1169366, -1118007, -888008, 1;
%t T[n_, k_]= Binomial[Binomial[n, 2] + k, k] + Binomial[Binomial[n, 2] + n-k, n-k] - Binomial[Binomial[n, 2] + n, n];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
%o (Magma)
%o T:= func< n,k | Binomial(Binomial(n,2) +k, k) + Binomial(Binomial(n,2) +n-k, n-k) - Binomial(Binomial(n,2) +n, n) >;
%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 09 2021
%o (Sage)
%o def f(n,k): return binomial(binomial(n,2) + k, k)
%o def T(n,k): return f(n,k) + f(n,n-k) - f(n,n)
%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 09 2021
%K sign,tabl
%O 0,8
%A _Roger L. Bagula_, Apr 22 2010
%E Edited by _G. C. Greubel_, Jul 09 2021
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