%I #35 Oct 04 2018 18:28:04

%S 1,1,1,1,1,1,1,2,1,1,1,6,4,1,1,1,24,36,8,1,1,1,120,576,216,16,1,1,1,

%T 720,14400,13824,1296,32,1,1,1,5040,518400,1728000,331776,7776,64,1,1,

%U 1,40320,25401600,373248000,207360000,7962624,46656,128,1,1

%N Square array A(n,k) = (k!)^n, n>=0, k>=0, read by antidiagonals.

%C A(n,k) is the determinant of the k X k matrix M = [Stirling2(n+i,j)] for 1<=i,j<=k. A(2,3) = det([1,3,1; 1,7,6; 1,15,25]) = 36.

%C A(n,k) is the determinant of the symmetric k X k matrix M = [sigma_n(gcd(i,j))] for 1<=i,j<=k. A(2,3) = det([1,1,1; 1,5,1; 1,1,10]) = 36.

%C A(n,k) is (-1)^(n*k) times the determinant of the n X n matrix M = [Stirling1(k+i,j)] for 1<=i,j<=n. A(2,3) = (-1)^(2+3) * det([-6,11; 24,-50]) = 36.

%C A(n,k) is the number of lattice paths from {n}^k to {0}^k using steps that decrement one component by 1 such that for each point (p_1,p_2,...,p_k) we have abs(p_i-p_j) <= 1 for 1<=i,j<=k. A(2,3) = 36:

%C (1,2,2)-(1,1,2) (0,1,1)-(0,0,1)

%C / X \ / X \

%C (2,2,2)-(2,1,2) (1,2,1)-(1,1,1)-(1,0,1) (0,1,0)-(0,0,0).

%C \ X / \ X /

%C (2,2,1) (2,1,1) (1,1,0) (1,0,0)

%C A(n,k) is the number of set partitions of [k*(n+1)] into k blocks of size n+1 such that the elements of each block are distinct mod n+1. A(2,3) = 36: 123|456|789, 126|345|789, ..., 189|234|567, 189|246|357.

%H Alois P. Heinz, <a href="/A225816/b225816.txt">Antidiagonals n = 0..36, flattened</a>

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

%F A(n,k) = k^n * A(n,k-1) for k>0, A(n,0) = 1.

%F A(n,k) = k! * A(n-1,k) for n>0, A(0,k) = 1.

%F G.f. of column k: 1/(1-k!*x).

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 2, 6, 24, 120, ...

%e 1, 1, 4, 36, 576, 14400, ...

%e 1, 1, 8, 216, 13824, 1728000, ...

%e 1, 1, 16, 1296, 331776, 207360000, ...

%e 1, 1, 32, 7776, 7962624, 24883200000, ...

%p A:= (n, k)-> k!^n:

%p seq(seq(A(n,d-n), n=0..d), d=0..12);

%Y Columns k=0+1, 2-4 give: A000012, A000079, A000400, A009968.

%Y Rows n=0-4 give: A000012, A000142, A001044, A000442, A134375.

%Y Main diagonal gives: A036740.

%Y Cf. A008275, A048994, A008277, A048993, A003989, A109004, A109974.

%K nonn,tabl

%O 0,8

%A _Alois P. Heinz_, Jul 29 2013