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A225816
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Square array A(n,k) = (k!)^n, n>=0, k>=0, read by antidiagonals.
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10
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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, 720, 14400, 13824, 1296, 32, 1, 1, 1, 5040, 518400, 1728000, 331776, 7776, 64, 1, 1, 1, 40320, 25401600, 373248000, 207360000, 7962624, 46656, 128, 1, 1
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OFFSET
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0,8
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COMMENTS
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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.
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.
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.
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:
(1,2,2)-(1,1,2) (0,1,1)-(0,0,1)
/ X \ / X \
(2,2,2)-(2,1,2) (1,2,1)-(1,1,1)-(1,0,1) (0,1,0)-(0,0,0).
\ X / \ X /
(2,2,1) (2,1,1) (1,1,0) (1,0,0)
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.
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LINKS
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FORMULA
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A(n,k) = (k!)^n.
A(n,k) = k^n * A(n,k-1) for k>0, A(n,0) = 1.
A(n,k) = k! * A(n-1,k) for n>0, A(0,k) = 1.
G.f. of column k: 1/(1-k!*x).
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 2, 6, 24, 120, ...
1, 1, 4, 36, 576, 14400, ...
1, 1, 8, 216, 13824, 1728000, ...
1, 1, 16, 1296, 331776, 207360000, ...
1, 1, 32, 7776, 7962624, 24883200000, ...
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MAPLE
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A:= (n, k)-> k!^n:
seq(seq(A(n, d-n), n=0..d), d=0..12);
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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