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%I #8 Jan 04 2022 02:21:43
%S 1,1,1,1,1,2,1,1,4,6,1,1,5,44,24,1,1,6,90,1144,120,1,1,7,162,5220,
%T 65208,720,1,1,8,266,18144,934380,7824960,5040,1,1,9,408,51604,
%U 8219232,507368340,1932765120,40320,1,1,10,594,126480,50313900,14942563776,830054604240,970248090240,362880
%N Array A(n, k) = Product_{j=1..n} ( j - (1+j)*(k+1) + (k+1)^(j+1) ) with A(n, 0) = n!, read by antidiagonals.
%H G. C. Greubel, <a href="/A156579/b156579.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = A(k, n-k) for the array defined by A(n, k) = Product_{j=1..n} ( j - (1+j)*(k+1) + (k+1)^(j+1) - 1 ) with A(n, 0) = n!.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 1, 2;
%e 1, 1, 4, 6;
%e 1, 1, 5, 44, 24;
%e 1, 1, 6, 90, 1144, 120;
%e 1, 1, 7, 162, 5220, 65208, 720;
%e 1, 1, 8, 266, 18144, 934380, 7824960, 5040;
%e 1, 1, 9, 408, 51604, 8219232, 507368340, 1932765120, 40320;
%t A[n_, k_]:= If[k==0, n!, k^(-2*n)*Product[j -(1+j)*(k+1) +(k+1)^(j+1), {j, n}] ];
%t T[n_, k_]:= A[k, n-k];
%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 04 2022 *)
%o (Sage)
%o def A(n,k): return factorial(n) if (k==0) else (1/k^(2*n))*product( j -(1+j)*(k+1) +(k+1)^(j+1) for j in (1..n) )
%o def T(n,k): return A(k, n-k)
%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jan 04 2022
%Y Cf. A156540.
%K nonn,tabl
%O 0,6
%A _Roger L. Bagula_, Feb 10 2009
%E Edited by _G. C. Greubel_, Jan 04 2022
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