login
Multifactorial array: A(k,n) = k-tuple factorial of n, for positive n, read by ascending antidiagonals.
3

%I #23 Nov 02 2021 22:25:48

%S 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,

%T 2,3,4,5,18,105,40320,1,2,3,4,5,12,28,384,362880,1,2,3,4,5,6,21,80,

%U 945,3628800,1,2,3,4,5,6,14,32,162,3840,39916800,1,2,3,4,5,6,7,24,45,280

%N Multifactorial array: A(k,n) = k-tuple factorial of n, for positive n, read by ascending antidiagonals.

%C 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.

%C Similar to A114423 (with rows and columns exchanged). - _Georg Fischer_, Nov 02 2021

%H Alois P. Heinz, <a href="/A129116/b129116.txt">Antidiagonals n = 1..141, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Multifactorial.html">Multifactorial.</a>

%F A(k,n) = n!k.

%F A(k,n) = M(n,k) in A114423. - _Georg Fischer_, Nov 02 2021

%e Table begins:

%e k / A(k,n)

%e 1.|.1.2.6.24.120.720.5040.40320.362880.3628800... = A000142.

%e 2.|.1.2.3..8..15..48..105...384....945....3840... = A006882.

%e 3.|.1.2.3..4..10..18...28....80....162.....280... = A007661.

%e 4.|.1.2.3..4...5..12...21....32.....45.....120... = A007662.

%e 5.|.1.2.3..4...5...6...14....24.....36......50... = A085157.

%e 6.|.1.2.3..4...5...6....7....16.....27......40... = A085158.

%p 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

%t 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_ *)

%Y Cf. A000142 (n!), A006882 (n!!), A007661 (n!!!), A007662(n!4), A085157 (n!5), A085158 (n!6), A114799 (n!7), A114800 (n!8), A114806 (n!9), A288327 (n!10).

%Y Cf. A114423 (transposed).

%K easy,nonn,tabl

%O 1,3

%A _Jonathan Vos Post_, May 24 2007

%E Corrected and extended by _Alois P. Heinz_, Feb 02 2009