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A353435
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Array read by descending antidiagonals: T(n,m) is the number of sequences of length n >= 0 with elements in 0..m-1 such that the Hankel matrix of any odd number of consecutive terms is invertible over the ring of integers modulo m >= 1.
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2
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 0, 1, 1, 4, 4, 4, 0, 1, 1, 2, 16, 0, 4, 0, 1, 1, 6, 4, 48, 0, 0, 0, 1, 1, 4, 36, 0, 144, 0, 0, 0, 1, 1, 6, 16, 180, 0, 320, 0, 0, 0, 1, 1, 4, 36, 0, 900, 0, 720, 0, 0, 0, 1, 1, 10, 16, 108, 0, 3744, 0, 1312, 0, 0, 0, 1
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OFFSET
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0,8
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COMMENTS
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T(n,m) is divisible by T(2,m) = A127473(n) for n >= 2, because if r and s are coprime to m, the sequence (x_1, ..., x_n) satisfies the conditions if and only if the sequence (r*s^0*x_1 mod m, ..., r*s^(n-1)*x_n mod m) does.
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LINKS
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FORMULA
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For fixed n, T(n,m) is multiplicative with T(n,p^e) = T(n,p)*p^(n*(e-1)).
T(n,m) = A353436(n,m) if m is prime.
T(3,m) = (m-1)^2*(m-2) = A045991(m-1) if m is prime.
T(4,m) = (m-1)^2*(m-2)^2 = A035287(m-1) if m is prime.
Empirically: T(5,m) = (m-1)^2*(m-3)*(m^2-4*m+5) if m >= 3 is prime.
T(n,2) = 0 for n >= 3.
T(n,3) = 0 for n >= 5.
T(n,5) = 0 for n >= 23.
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EXAMPLE
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Array begins:
n\m| 1 2 3 4 5 6 7 8 9 10
---+--------------------------------------
0 | 1 1 1 1 1 1 1 1 1 1
1 | 1 1 2 2 4 2 6 4 6 4
2 | 1 1 4 4 16 4 36 16 36 16
3 | 1 0 4 0 48 0 180 0 108 0
4 | 1 0 4 0 144 0 900 0 324 0
5 | 1 0 0 0 320 0 3744 0 0 0
6 | 1 0 0 0 720 0 15552 0 0 0
7 | 1 0 0 0 1312 0 54216 0 0 0
8 | 1 0 0 0 2400 0 189468 0 0 0
9 | 1 0 0 0 3232 0 550728 0 0 0
10 | 1 0 0 0 4560 0 1604088 0 0 0
11 | 1 0 0 0 4656 0 3895560 0 0 0
12 | 1 0 0 0 4928 0 9467856 0 0 0
13 | 1 0 0 0 4368 0 19185516 0 0 0
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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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