cp's OEIS Frontend

A129116 Multifactorial array: A(k,n) = k-tuple factorial of n, for positive n, read by ascending antidiagonals.

%I A129116 #30 Aug 21 2025 05:52:33
%S A129116 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 A129116 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 A129116 945,3628800,1,2,3,4,5,6,14,32,162,3840,39916800,1,2,3,4,5,6,7,24,45,280,10395,479001600
%N A129116 Multifactorial array: A(k,n) = k-tuple factorial of n, for positive n, read by ascending antidiagonals.
%C A129116 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 A129116 Similar to A114423 (with rows and columns exchanged). - _Georg Fischer_, Nov 02 2021
%H A129116 Alois P. Heinz, <a href="/A129116/b129116.txt">Antidiagonals n = 1..141, flattened</a>
%H A129116 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Multifactorial.html">Multifactorial.</a>
%F A129116 A(k,n) = n!k.
%F A129116 A(k,n) = M(n,k) in A114423. - _Georg Fischer_, Nov 02 2021
%e A129116 Table begins:
%e A129116   k / A(k,n)
%e A129116   1 | 1 2 6 24 120 720 5040 40320 362880 3628800 ... = A000142.
%e A129116   2 | 1 2 3  8  15  48  105   384    945    3840 ... = A006882.
%e A129116   3 | 1 2 3  4  10  18   28    80    162     280 ... = A007661.
%e A129116   4 | 1 2 3  4   5  12   21    32     45     120 ... = A007662.
%e A129116   5 | 1 2 3  4   5   6   14    24     36      50 ... = A085157.
%e A129116   6 | 1 2 3  4   5   6    7    16     27      40 ... = A085158.
%p A129116 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 A129116 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 A129116 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 A129116 Cf. A114423 (transposed).
%K A129116 easy,nonn,tabl
%O A129116 1,3
%A A129116 _Jonathan Vos Post_, May 24 2007
%E A129116 Corrected and extended by _Alois P. Heinz_, Feb 02 2009