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A156953
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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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4
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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, 193786425, 720, 1, 1, 7, 1116, 4685625, 108886835200, 119216439727875, 5040, 1, 1, 8, 2107, 32381856, 14260348265625, 9975288480661504000, 9314352420075537699375, 40320
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OFFSET
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0,6
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REFERENCES
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Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 182.
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LINKS
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FORMULA
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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!.
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
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EXAMPLE
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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, 193786425, 720;
1, 1, 7, 1116, 4685625, 108886835200, 119216439727875, 5040;
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MATHEMATICA
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A[n_, k_]:= If[k==0, n!, Product[QPochhammer[k+1, k+1, j]/(-k)^j, {j, n}]];
T[n_, k_]:= A[k, n-k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 08 2022 *)
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PROG
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(Sage)
from sage.combinat.q_analogues import q_pochhammer
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) )
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 08 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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