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Array A(n, k) = Product_{j=1..n} Product_{i=1..j} (1 - (k+1)^i)/(1 - (k+1))^j, with A(n, k) = n!, read by antidiagonals.
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%I #11 Jan 09 2022 09:57:44

%S 1,1,1,1,1,2,1,1,3,6,1,1,4,63,24,1,1,5,208,19845,120,1,1,6,525,432640,

%T 193786425,720,1,1,7,1116,4685625,108886835200,119216439727875,5040,1,

%U 1,8,2107,32381856,14260348265625,9975288480661504000,9314352420075537699375,40320

%N Array A(n, k) = Product_{j=1..n} Product_{i=1..j} (1 - (k+1)^i)/(1 - (k+1))^j, with A(n, k) = n!, read by antidiagonals.

%D Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 182.

%H G. C. Greubel, <a href="/A156953/b156953.txt">Rows n = 0..20 of the triangle, flattened</a>

%F T(n, k) = A(k, n-k) where the array is given by A(n, k) = Product_{j=1..n} Product_{i=1..j} (1 - (k+1)^i)/(1 - (k+1))^j, with A(n, k) = n!.

%F T(n, k) = Product_{j=1..k} (q-Pochhammer(j, n-k+1, n-k+1)/(-k)^j) with T(n, n) = n!. - _G. C. Greubel_, Jan 08 2022

%e 1;

%e 1, 1;

%e 1, 1, 2;

%e 1, 1, 3, 6;

%e 1, 1, 4, 63, 24;

%e 1, 1, 5, 208, 19845, 120;

%e 1, 1, 6, 525, 432640, 193786425, 720;

%e 1, 1, 7, 1116, 4685625, 108886835200, 119216439727875, 5040;

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

%t T[n_, k_]:= A[k, n-k];

%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 08 2022 *)

%o (Sage)

%o from sage.combinat.q_analogues import q_pochhammer

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

%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jan 08 2022

%Y Cf. A156950, A156951, A156952.

%K nonn,tabl

%O 0,6

%A _Roger L. Bagula_, Feb 19 2009

%E Edited by _G. C. Greubel_, Jan 08 2022