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Array A(n, k) = Product_{j=1..n} ( (k+1)^j - 1 ) with A(n, 0) = n!, read by antidiagonals.
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%I #12 Jan 04 2022 02:21:48

%S 1,1,1,1,1,2,1,1,3,6,1,1,4,21,24,1,1,5,52,315,120,1,1,6,105,2080,9765,

%T 720,1,1,7,186,8925,251680,615195,5040,1,1,8,301,29016,3043425,

%U 91611520,78129765,40320,1,1,9,456,77959,22661496,4154275125,100131391360,19923090075,362880

%N Array A(n, k) = Product_{j=1..n} ( (k+1)^j - 1 ) with A(n, 0) = n!, read by antidiagonals.

%H G. C. Greubel, <a href="/A156540/b156540.txt">antidiagonals n = 0..50, flattened</a>

%F T(n, k) = A(k, n-k) for the array defined by A(n, k) = Product_{j=1..n} ( (k+1)^j - 1 ) with A(n, 0) = n!.

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 1, 2;

%e 1, 1, 3, 6;

%e 1, 1, 4, 21, 24;

%e 1, 1, 5, 52, 315, 120;

%e 1, 1, 6, 105, 2080, 9765, 720;

%e 1, 1, 7, 186, 8925, 251680, 615195, 5040;

%e 1, 1, 8, 301, 29016, 3043425, 91611520, 78129765, 40320;

%e 1, 1, 9, 456, 77959, 22661496, 4154275125, 100131391360, 19923090075, 362880;

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

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

%t Table[T[n, k], {n,0,12}, {k,0,n}]//TableForm (* modified by _G. C. Greubel_, Jan 04 2022 *)

%o (Sage)

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

%o def T(n,k): return A(k, n-k)

%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jan 04 2022

%Y Cf. A156579.

%K nonn,tabl

%O 0,6

%A _Roger L. Bagula_, Feb 09 2009

%E Edited by _G. C. Greubel_, Jan 04 2022