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%I #11 Jul 17 2023 01:04:30
%S 1,6,1,480,2880,1,21840,2882880,18162144000,40040,626400,473558400,
%T 3270820512960000,145032891526185062400000,380331009246988800000,
%U 14968800,41254012800,22288874800832640000,17065364402126896882035609600000,59861520269419187714435515882890362880000000,687565882176828511388211047069939545826918400000000
%N Triangle T(n, k) = (n+2)!*(n*(n+1)*(2*n+1)/6)!/( (k^2)! * abs(2 + 2*k^2 - (n*(n + 1)*(2*n+1)/6))! ), read by rows.
%H G. C. Greubel, <a href="/A123147/b123147.txt">Rows n = 0..15 of the triangle, flattened</a>
%F T(n, k) = (n+2)!*(n*(n+1)*(2*n+1)/6)!/( (k^2)! * abs(2 + 2*k^2 - (n*(n + 1)*(2*n+1)/6))! ).
%e Triangle begins as:
%e 1;
%e 6, 1;
%e 480, 2880, 1;
%e 21840, 2882880, 18162144000, 40040;
%t T[n_, k_]= (n+2)!*(n*(n+1)*(2*n+1)/6)!/((k^2)!*Abs[2 +2*k^2 -(n*(n + 1)*(2*n+1)/6)]!);
%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
%o (Magma)
%o F:=Factorial;
%o A123147:= func< n,k | F(n+2)*F(Floor(n*(n+1)*(2*n+1)/6))/( F(k^2) * F(Abs(Floor(2 + 2*k^2 - n*(n+1)*(2*n+1)/6))) ) >;
%o [A123147(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Jul 16 2023
%o (SageMath)
%o f=factorial
%o def A123147(n, k): return f(n+2)*f(n*(n+1)*(2*n+1)/6)/(f(k^2)*f(abs(2 + 2*k^2 - (n*(n+1)*(2*n+1)/6))) )
%o flatten([[A123147(n,k) for k in range(n+1)] for n in range(11)]) # _G. C. Greubel_, Jul 16 2023
%Y Cf. A123146.
%K nonn,tabl,easy
%O 0,2
%A _Roger L. Bagula_, Oct 01 2006
%E Edited by _G. C. Greubel_, Jul 16 2023
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