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Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ( (k+1)^7 - Sum_{m=1..5} (k+1)^m )^i ) with T(n, 0) = n!, read by antidiagonals.
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%I #11 Jun 14 2021 18:12:06

%S 1,1,1,1,1,2,1,1,67,6,1,1,1825,296341,24,1,1,15021,6075061825,

%T 86507568379,120,1,1,74221,3388969238841,36886153511769270625,

%U 1666711474847102245,720,1,1,270607,408859932813241,11484347898092358710495061,408508286631392559053881969770625,2119389029714451057320373595,5040

%N Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ( (k+1)^7 - Sum_{m=1..5} (k+1)^m )^i ) with T(n, 0) = n!, read by antidiagonals.

%H G. C. Greubel, <a href="/A156888/b156888.txt">Antidiagonal rows n = 0..20, flattened</a>

%F T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ( (k+1)^7 - Sum_{m=1..5} (k+1)^m )^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+1)/k)*((k+1)^7 - (k+1)^6 - (k+1)^5 + 1) (square array). - _G. C. Greubel_, Jun 14 2021

%e Square array begins as:

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

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

%e 2, 67, 1825, ...;

%e 6, 296341, 6075061825, ...;

%e 24, 86507568379, 36886153511769270625, ...;

%e Antidiagonal triangle begins as:

%e 1;

%e 1, 1;

%e 1, 1, 2;

%e 1, 1, 67, 6;

%e 1, 1, 1825, 296341, 24;

%e 1, 1, 15021, 6075061825, 86507568379, 120;

%e 1, 1, 74221, 3388969238841, 36886153511769270625, 1666711474847102245, 720; ...

%t (* First program *)

%t T[n_, m_] = If[m==0, n!, Product[Sum[((m+1)^7 - Sum[(m+1)^r, {r,1,5}])^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+1)/n)*((n+1)^7 - (n+1)^6 - (n+1)^5 + 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+1)/n)*((n+1)^7 - (n+1)^6 - (n+1)^5 + 1)

%o def A156888(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([[A156888(k,n-k) for k in (0..n)] for n in (0..6)]) # _G. C. Greubel_, Jun 14 2021

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