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 A129116 Multifactorial array A[k,n] = k-tuple factorial of n, for positive n, by antidiagonals. 2
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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. LINKS Alois P. Heinz, Antidiagonals n = 1..141, flattened Eric Weisstein's World of Mathematics, Multifactorial. FORMULA A[k,n] = n!k, by antidiagonals. EXAMPLE 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. MAPLE 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 MATHEMATICA 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 *) CROSSREFS Cf. A000142, A006882, A007661, A007662, A085157, A085158. Sequence in context: A178803 A292901 A083773 * A096179 A166350 A210227 Adjacent sequences:  A129113 A129114 A129115 * A129117 A129118 A129119 KEYWORD easy,nonn,tabl AUTHOR Jonathan Vos Post, May 24 2007 EXTENSIONS Corrected and extended by Alois P. Heinz, Feb 02 2009 STATUS approved

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Last modified March 30 10:09 EDT 2020. Contains 333125 sequences. (Running on oeis4.)