

A129116


Multifactorial array A[k,n] = ktuple 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
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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, nk) elif n >= 1k then 1 else 0 fi end: seq(seq(A(1+dn, 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, nk], If[n >= 1k, 1, 0]]; Table[ A[1+dn, n], {d, 1, 16}, {n, 1, d}] // Flatten (* JeanFrançois 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



