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Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^2 - (k+1))^i ) with T(n, 0) = n!, read by antidiagonals.
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%I #16 Jun 14 2021 03:07:13

%S 1,1,1,1,1,2,1,1,3,6,1,1,7,21,24,1,1,13,301,315,120,1,1,21,2041,77959,

%T 9765,720,1,1,31,8841,3847285,121226245,615195,5040,1,1,43,28861,

%U 74450061,87029433985,1131162092095,78129765,40320,1,1,57,77701,806116591,12538953723681,23624400943530205,63330372050122765,19923090075,362880

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

%H G. C. Greubel, <a href="/A156881/b156881.txt">Antidiagonal rows n = 0..25, flattened</a>

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

%F T(n, k) = ( Product_{j=1..n} (k^j*(k+1)^j -1) )/(k^2 + k -1)^n with T(n, 0) = n! (square array). - _G. C. Greubel_, Jun 12 2021

%e Square array begins as:

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

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

%e 2, 3, 7, 13, 21, ...;

%e 6, 21, 301, 2041, 8841, ...;

%e 24, 315, 77959, 3847285, 74450061, ...;

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 1, 2;

%e 1, 1, 3, 6;

%e 1, 1, 7, 21, 24;

%e 1, 1, 13, 301, 315, 120;

%e 1, 1, 21, 2041, 77959, 9765, 720;

%e 1, 1, 31, 8841, 3847285, 121226245, 615195, 5040;

%e 1, 1, 43, 28861, 74450061, 87029433985, 1131162092095, 78129765, 40320;

%t (* First program *)

%t T[n_, m_] = If[m==0, n!, Product[Sum[(-(m+1) + (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 12 2021 *)

%t (* Second program *)

%t T[n_, k_]= If[k==0, n!, Product[(k^j*(k+1)^j -1), {j,n}]/(k^2+k-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 A156881(n, k): return factorial(n) if (k==0) else product((k^j*(k+1)^j -1) for j in (1..n))/(k^2 +k-1)^n

%o flatten([[A156881(k,n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 12 2021

%Y Cf. A156882, A156883.

%K nonn,tabl

%O 0,6

%A _Roger L. Bagula_, Feb 17 2009

%E Edited by _G. C. Greubel_, Jun 12 2021