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A359615 a(n) is the maximal determinant of an n X n Hermitian Toeplitz matrix using all the integers 1, 2, ..., n and with all off-diagonal elements purely imaginary.

%I A359615 #23 Jan 25 2023 20:50:17
%S A359615 1,1,3,9,512,9195,242931,7459494,524426191,17012915860,773407040859
%N A359615 a(n) is the maximal determinant of an n X n Hermitian Toeplitz matrix using all the integers 1, 2, ..., n and with all off-diagonal elements purely imaginary.
%H A359615 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>.
%e A359615 a(4) = 512:
%e A359615   [   1,  4*i,  2*i, 3*i;
%e A359615    -4*i,    1,  4*i, 2*i;
%e A359615    -2*i, -4*i,    1, 4*i;
%e A359615    -3*i, -2*i, -4*i,   1 ]
%t A359615 a={1}; For[n=1, n<=8, n++, mx=-Infinity; For[d=1, d<=n, d++, For[i=1, i<=(n-1)!, i++, If[(t=Det[ToeplitzMatrix[Join[{d}, I Part[Permutations[Drop[Range[n], {d}]], i]]]])>mx, mx=t]]]; AppendTo[a, mx]]; a
%o A359615 (Python)
%o A359615 from itertools import permutations
%o A359615 from sympy import Matrix, I
%o A359615 def A359615(n): return max(Matrix(n,n,[(d[i-j] if i>j else -d[j-i]) if i!=j else d[0]*I for i in range(n) for j in range(n)]).det()*(1,-I,-1,I)[n&3] for d in permutations(range(1,n+1))) # _Chai Wah Wu_, Jan 25 2023
%Y A359615 Cf. A350954, A359559, A359561.
%Y A359615 Cf. A359614 (minimal), A359616 (minimal permanent), A359617 (maximal permanent).
%K A359615 nonn,hard,more
%O A359615 0,3
%A A359615 _Stefano Spezia_, Jan 07 2023