login
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.
1

%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