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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.
2

%I #12 Apr 26 2022 10:23:23

%S 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,

%T 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,

%U 720,0,0,0,1,1,10,16,108,0,3744,0,1312,0,0,0,1

%N 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.

%C 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.

%F For fixed n, T(n,m) is multiplicative with T(n,p^e) = T(n,p)*p^(n*(e-1)).

%F T(n,m) = A353436(n,m) if m is prime.

%F T(3,m) = (m-1)^2*(m-2) = A045991(m-1) if m is prime.

%F T(4,m) = (m-1)^2*(m-2)^2 = A035287(m-1) if m is prime.

%F Empirically: T(5,m) = (m-1)^2*(m-3)*(m^2-4*m+5) if m >= 3 is prime.

%F T(n,2) = 0 for n >= 3.

%F T(n,3) = 0 for n >= 5.

%F T(n,5) = 0 for n >= 23.

%e Array begins:

%e n\m| 1 2 3 4 5 6 7 8 9 10

%e ---+--------------------------------------

%e 0 | 1 1 1 1 1 1 1 1 1 1

%e 1 | 1 1 2 2 4 2 6 4 6 4

%e 2 | 1 1 4 4 16 4 36 16 36 16

%e 3 | 1 0 4 0 48 0 180 0 108 0

%e 4 | 1 0 4 0 144 0 900 0 324 0

%e 5 | 1 0 0 0 320 0 3744 0 0 0

%e 6 | 1 0 0 0 720 0 15552 0 0 0

%e 7 | 1 0 0 0 1312 0 54216 0 0 0

%e 8 | 1 0 0 0 2400 0 189468 0 0 0

%e 9 | 1 0 0 0 3232 0 550728 0 0 0

%e 10 | 1 0 0 0 4560 0 1604088 0 0 0

%e 11 | 1 0 0 0 4656 0 3895560 0 0 0

%e 12 | 1 0 0 0 4928 0 9467856 0 0 0

%e 13 | 1 0 0 0 4368 0 19185516 0 0 0

%Y Cf. A035287, A045991, A350364, A353433, A353436.

%Y Rows: A000012 (n=0), A000010 (n=1), A127473 (n=2).

%Y Columns: A000012 (m=1), A130716 (m=2), A166926 (m=4 and m=6).

%K nonn,tabl

%O 0,8

%A _Pontus von Brömssen_, Apr 21 2022