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A129116
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Multifactorial array: A(k,n) = k-tuple factorial of n, for positive n, read by ascending antidiagonals.
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3
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1, 1, 2, 1, 2, 6, 1, 2, 3, 24, 1, 2, 3, 8, 120, 1, 2, 3, 4, 15, 720, 1, 2, 3, 4, 10, 48, 5040, 1, 2, 3, 4, 5, 18, 105, 40320, 1, 2, 3, 4, 5, 12, 28, 384, 362880, 1, 2, 3, 4, 5, 6, 21, 80, 945, 3628800, 1, 2, 3, 4, 5, 6, 14, 32, 162, 3840, 39916800, 1, 2, 3, 4, 5, 6, 7, 24, 45, 280
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OFFSET
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1,3
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COMMENTS
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The term "Quintuple factorial numbers" is also used for the sequences A008546, A008548, A052562, A047055, A047056 which have a different definition. The definition given here is the one commonly used. This problem exists for the other rows as well. "n!2" = n!!, "n!3" = n!!!, "n!4" = n!!!!, etcetera. Main diagonal is A[n,n] = n!n = n.
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LINKS
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FORMULA
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A(k,n) = n!k.
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EXAMPLE
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Table begins:
k / A(k,n)
1.|.1.2.6.24.120.720.5040.40320.362880.3628800... = A000142.
2.|.1.2.3..8..15..48..105...384....945....3840... = A006882.
3.|.1.2.3..4..10..18...28....80....162.....280... = A007661.
4.|.1.2.3..4...5..12...21....32.....45.....120... = A007662.
5.|.1.2.3..4...5...6...14....24.....36......50... = A085157.
6.|.1.2.3..4...5...6....7....16.....27......40... = A085158.
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MAPLE
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A:= proc(k, n) option remember; if n >= 1 then n* A(k, n-k) elif n >= 1-k then 1 else 0 fi end: seq(seq(A(1+d-n, n), n=1..d), d=1..16); # Alois P. Heinz, Feb 02 2009
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MATHEMATICA
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A[k_, n_] := A[k, n] = If[n >= 1, n*A[k, n-k], If[n >= 1-k, 1, 0]]; Table[ A[1+d-n, n], {d, 1, 16}, {n, 1, d}] // Flatten (* Jean-François Alcover, May 27 2016, after Alois P. Heinz *)
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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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