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Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^3 - (k+1))^i ) with T(n, 0) = n!, read by antidiagonals.
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%I #15 Jun 14 2021 17:51:56

%S 1,1,1,1,1,2,1,1,7,6,1,1,25,301,24,1,1,61,15025,77959,120,1,1,121,

%T 223321,216735625,121226245,720,1,1,211,1757041,49054914181,

%U 75034090110625,1131162092095,5040,1,1,337,9349621,3061680840361,646527139289672641,623445123763413765625,63330372050122765,40320

%N Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^3 - (k+1))^i ) with T(n, 0) = n!, read by antidiagonals.

%H G. C. Greubel, <a href="/A156882/b156882.txt">Antidiagonal rows n = 0..25, flattened</a>

%F T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^3 - (k+1))^i ) with T(n, 0) = n! (square array).

%F T(n, k) = ( Product_{j=1..n} ((k*(k+1)*(k+2))^j -1) )/(k*(k+1)*(k+2) -1)^n with T(n, 0) = n! (square array). - _G. C. Greubel_, Jun 14 2021

%e Square array begins as:

%e 1, 1, 1, 1, 1, 1 ...;

%e 1, 1, 1, 1, 1, 1 ...;

%e 2, 7, 25, 61, 121, 211 ...;

%e 6, 301, 15025, 223321, 1757041, 9349621 ...;

%e 24, 77959, 216735625, 49054914181, 3061680840361, 87001131137131 ...;

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 1, 2;

%e 1, 1, 7, 6;

%e 1, 1, 25, 301, 24;

%e 1, 1, 61, 15025, 77959, 120;

%e 1, 1, 121, 223321, 216735625, 121226245, 720;

%e 1, 1, 211, 1757041, 49054914181, 75034090110625, 1131162092095, 5040;

%t (* First program *)

%t T[n_, m_] = If[m==0, n!, Product[Sum[(-(m+1) + (m+1)^3)^i, {i,0,k-1}], {k,n}]];

%t Table[T[k,n-k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jun 12 2021 *)

%t (* Second program *)

%t T[n_, k_]= If[k==0, n!, Product[((k*(k+1)*(k+2))^j -1), {j,n}]/(k*(k+1)*(k+2) - 1)^n];

%t Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 14 2021 *)

%o (Sage)

%o def A156882(n, k): return factorial(n) if (k==0) else product(((k*(k+1)*(k+2))^j -1) for j in (1..n))/(k*(k+1)*(k+2)-1)^n

%o flatten([[A156882(k,n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 14 2021

%Y Cf. A156881, A156883, A156885, A156888, A156889.

%K nonn,tabl

%O 0,6

%A _Roger L. Bagula_, Feb 17 2009

%E Edited by _G. C. Greubel_, Jun 14 2021