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Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^5 - (k+1)^4 - (k+1)^3 + (k+1)^2)^i ) with T(n, 0) = n!, read by antidiagonals.
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%I #14 Jun 14 2021 15:59:58

%S 1,1,1,1,1,2,1,1,13,6,1,1,145,2041,24,1,1,721,3027745,3847285,120,1,1,

%T 2401,374286241,9104020469425,87029433985,720,1,1,6301,13835524801,

%U 139895890728482161,3941936722370875247425,23624400943530205,5040,1,1

%N Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^5 - (k+1)^4 - (k+1)^3 + (k+1)^2)^i ) with T(n, 0) = n!, read by antidiagonals.

%H G. C. Greubel, <a href="/A156885/b156885.txt">Antidiagonal rows n = 0..25, flattened</a>

%F T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^5 - (k+1)^4 - (k+1)^3 + (k+1)^2)^i ) with T(n, 0) = n! (square array).

%F T(n, k) = ( Product_{j=1..n} ((k^2*(k+1)^2*(k+2))^j -1) )/(k^2*(k+1)^2*(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, ...;

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

%e 2, 13, 145, 721, ...;

%e 6, 2041, 3027745, 374286241, ...;

%e 24, 3847285, 9104020469425, 139895890728482161, ...;

%e Antidiagonal triangle begins as:

%e 1;

%e 1, 1;

%e 1, 1, 2;

%e 1, 1, 13, 6;

%e 1, 1, 145, 2041, 24;

%e 1, 1, 721, 3027745, 3847285, 120;

%e 1, 1, 2401, 374286241, 9104020469425, 87029433985, 720; ...

%t (* First program *)

%t T[n_, m_] = If[m==0, n!, Product[Sum[((m+1)^5 -(m+1)^4 -(m+1)^3 +(m+1)^2)^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 14 2021 *)

%t (* Second program *)

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

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

%o (Sage)

%o def A156885(n, k): return factorial(n) if (k==0) else product(( (k^2*(k+1)^2*(k+2))^j -1) for j in (1..n))/(k^2*(k+1)^2*(k+2) -1)^n

%o flatten([[A156885(k,n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 14 2021

%Y Cf. A156881, A156882, A156883, A156888, A156889.

%K nonn,tabl

%O 0,6

%A _Roger L. Bagula_, Feb 17 2009

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