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A385034 Absolute value squared of the maximal determinant of a matrix of order n with entries in the third roots of unity.

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%I A385034 #34 Jun 25 2025 06:39:37
%S A385034 1,1,3,27,189,1701,46656,606528,8957952,387420489,7360989291,
%T A385034 154580775111,8916100448256,222902511206400
%N A385034 Absolute value squared of the maximal determinant of a matrix of order n with entries in the third roots of unity.
%C A385034 The n-th term is equal to n^n if and only if a Butson-type Hadamard matrix over the third roots exist.
%H A385034 G. Nuñez Ponasso, <a href="https://doi.org/10.1016/j.laa.2025.05.024">Maximal determinants of matrices over the roots of unity</a>, Linear Algebra and its Applications, 723 (2025), 201-243.
%e A385034 With w=exp(2*Pi*i/3) and i the imaginary unit, a matrix corresponding to a(3) = 27 is
%e A385034   [1, 1,   1;
%e A385034    1, w,   w^2;
%e A385034    1, w^2, w],
%e A385034 and a matrix corresponding to a(4) = 189 is
%e A385034   [w, 1, 1, 1;
%e A385034    1, w, 1, 1;
%e A385034    1, 1, w, 1;
%e A385034    1, 1, 1, w].
%Y A385034 Cf. A000312, A003433.
%K A385034 nonn,hard,more
%O A385034 0,3
%A A385034 _Guillermo N. Ponasso_, Jun 16 2025