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Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ( (k+1)^6 -(k+1)^5 -(k+1)^4 +(k+1)^2 )^i ) with T(n, 0) = n!, read by antidiagonals.
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%I #11 Jun 14 2021 18:11:43

%S 1,1,1,1,1,2,1,1,21,6,1,1,415,8841,24,1,1,2833,71301565,74450061,120,

%T 1,1,11901,22729320481,5071662849566575,12538953723681,720,1,1,37621,

%U 1685442243801,516439650916945061425,149348900281032409928364325,42236475040875277701,5040

%N Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ( (k+1)^6 -(k+1)^5 -(k+1)^4 +(k+1)^2 )^i ) with T(n, 0) = n!, read by antidiagonals.

%H G. C. Greubel, <a href="/A156889/b156889.txt">Antidiagonal rows n = 0..25, flattened</a>

%F T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ( (k+1)^6 -(k+1)^5 -(k+1)^4 +(k+1)^2 )^i ) with T(n, 0) = n! (square array).

%F T(n, k) = ( Product_{j=1..n} (f(k)^j -1) )/(f(k) -1)^n with T(n, 0) = n! and f(k) = k*(k+1)^2*(k^3 -3*k^2 -2*k -1) (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, 21, 415, 2833, ...;

%e 6, 8841, 71301565, 22729320481, ...;

%e 24, 74450061, 5071662849566575, 516439650916945061425, ...;

%e Antidiagonal triangle begins as:

%e 1;

%e 1, 1;

%e 1, 1, 2;

%e 1, 1, 21, 6;

%e 1, 1, 415, 8841, 24;

%e 1, 1, 2833, 71301565, 74450061, 120;

%e 1, 1, 11901, 22729320481, 5071662849566575, 12538953723681, 720; ...

%t (* First program *)

%t T[n_, m_] = If[m==0, n!, Product[Sum[((m+1)^6 -(m+1)^5 -(m+1)^4 +(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 f[n_]:= n*(n+1)^2*(n^3 +3*n^2 +2*n -1);

%t T[n_, k_]= If[k==0, n!, Product[(f[k]^j -1), {j,n}]/(f[k]-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 f(n): return n*(n+1)^2*(n^3 +3*n^2 +2*n -1)

%o def A156889(n, k): return factorial(n) if (k==0) else product( (f(k)^j - 1) for j in (1..n))/( f(k) -1 )^n

%o flatten([[A156889(k,n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 14 2021

%Y Cf. A156881, A156882, A156883, A156885, A156888.

%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