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A156540
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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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2
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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, 720, 1, 1, 7, 186, 8925, 251680, 615195, 5040, 1, 1, 8, 301, 29016, 3043425, 91611520, 78129765, 40320, 1, 1, 9, 456, 77959, 22661496, 4154275125, 100131391360, 19923090075, 362880
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OFFSET
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0,6
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LINKS
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FORMULA
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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!.
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EXAMPLE
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Triangle begins as:
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, 720;
1, 1, 7, 186, 8925, 251680, 615195, 5040;
1, 1, 8, 301, 29016, 3043425, 91611520, 78129765, 40320;
1, 1, 9, 456, 77959, 22661496, 4154275125, 100131391360, 19923090075, 362880;
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MATHEMATICA
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A[n_, k_]:= If[k==0, n!, (1/k^n)*Product[ (k + 1)^j -1, {j, n}] ];
T[n_, k_]:= A[k, n-k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//TableForm (* modified by G. C. Greubel, Jan 04 2022 *)
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PROG
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(Sage)
def A(n, k): return factorial(n) if (k==0) else (1/k^n)*product( (k+1)^j -1 for j in (1..n) )
def T(n, k): return A(k, n-k)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 04 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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