cp's OEIS Frontend

A353435 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.

%I A353435 #12 Apr 26 2022 10:23:23
%S A353435 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 A353435 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 A353435 720,0,0,0,1,1,10,16,108,0,3744,0,1312,0,0,0,1
%N A353435 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 A353435 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 A353435 For fixed n, T(n,m) is multiplicative with T(n,p^e) = T(n,p)*p^(n*(e-1)).
%F A353435 T(n,m) = A353436(n,m) if m is prime.
%F A353435 T(3,m) = (m-1)^2*(m-2) = A045991(m-1) if m is prime.
%F A353435 T(4,m) = (m-1)^2*(m-2)^2 = A035287(m-1) if m is prime.
%F A353435 Empirically: T(5,m) = (m-1)^2*(m-3)*(m^2-4*m+5) if m >= 3 is prime.
%F A353435 T(n,2) = 0 for n >= 3.
%F A353435 T(n,3) = 0 for n >= 5.
%F A353435 T(n,5) = 0 for n >= 23.
%e A353435 Array begins:
%e A353435   n\m| 1  2  3  4    5  6        7  8   9 10
%e A353435   ---+--------------------------------------
%e A353435    0 | 1  1  1  1    1  1        1  1   1  1
%e A353435    1 | 1  1  2  2    4  2        6  4   6  4
%e A353435    2 | 1  1  4  4   16  4       36 16  36 16
%e A353435    3 | 1  0  4  0   48  0      180  0 108  0
%e A353435    4 | 1  0  4  0  144  0      900  0 324  0
%e A353435    5 | 1  0  0  0  320  0     3744  0   0  0
%e A353435    6 | 1  0  0  0  720  0    15552  0   0  0
%e A353435    7 | 1  0  0  0 1312  0    54216  0   0  0
%e A353435    8 | 1  0  0  0 2400  0   189468  0   0  0
%e A353435    9 | 1  0  0  0 3232  0   550728  0   0  0
%e A353435   10 | 1  0  0  0 4560  0  1604088  0   0  0
%e A353435   11 | 1  0  0  0 4656  0  3895560  0   0  0
%e A353435   12 | 1  0  0  0 4928  0  9467856  0   0  0
%e A353435   13 | 1  0  0  0 4368  0 19185516  0   0  0
%Y A353435 Cf. A035287, A045991, A350364, A353433, A353436.
%Y A353435 Rows: A000012 (n=0), A000010 (n=1), A127473 (n=2).
%Y A353435 Columns: A000012 (m=1), A130716 (m=2), A166926 (m=4 and m=6).
%K A353435 nonn,tabl
%O A353435 0,8
%A A353435 _Pontus von Brömssen_, Apr 21 2022