OFFSET
0,9
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
FORMULA
A(n,k) = Product_{i=0..k-1} floor((n+i)/k)!.
A(k*n,k) = (n!)^k = A225816(k,n).
For k > 0, A(n, k) ~ (2*Pi*n)^((k - 1)/2) * n! / k^(n + k/2). - Vaclav Kotesovec, Oct 02 2018
EXAMPLE
A(5,0) = A(5,5) = 1: 12345.
A(5,1) = 5! = 120: all permutations of {1,2,3,4,5}.
A(5,2) = 12: 12345, 12543, 14325, 14523, 32145, 32541, 34125, 34521, 52143, 52341, 54123, 54321.
A(5,3) = 4: 12345, 15342, 42315, 45312.
A(5,4) = 2: 12345, 52341.
A(7,4) = 8: 1234567, 1274563, 1634527, 1674523, 5234167, 5274163, 5634127, 5674123.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 6, 2, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 24, 4, 2, 1, 1, 1, 1, 1, 1, 1, ...
1, 120, 12, 4, 2, 1, 1, 1, 1, 1, 1, ...
1, 720, 36, 8, 4, 2, 1, 1, 1, 1, 1, ...
1, 5040, 144, 24, 8, 4, 2, 1, 1, 1, 1, ...
1, 40320, 576, 72, 16, 8, 4, 2, 1, 1, 1, ...
1, 362880, 2880, 216, 48, 16, 8, 4, 2, 1, 1, ...
1, 3628800, 14400, 864, 144, 32, 16, 8, 4, 2, 1, ...
MAPLE
A:= (n, k)-> mul(floor((n+i)/k)!, i=0..k-1):
seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
A[n_, k_] := Product[Floor[(n+i)/k]!, {i, 0, k-1}];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 26 2019, from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 15 2016
STATUS
approved