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%I #22 Jan 25 2023 18:37:08
%S 1,1,-3,-16,-36,-40,20,184,400,432,-112,-1472,-3136,-3328,576,9856,
%T 20736,21760,-2816,-59392,-123904,-129024,13312,333824,692224,716800,
%U -61440,-1785856,-3686400,-3801088,278528,9207808,18939904,19464192,-1245184,-46137344,-94633984
%N a(n) is the determinant of an n X n Hermitian Toeplitz matrix whose first row consists of 1, 2*i, ..., n*i, where i denotes the imaginary unit.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%F A359614(n) <= a(n) <= A359615(n).
%F Conjectured formulas: (Start)
%F O.g.f.: (1 - 5*x + 9*x^2 - 12*x^3 + 10*x^4 - 4*x^5)/(1 - 2*x + 2*x^2)^3.
%F a(n) = 6*a(n-1) - 18*a(n-2) + 32*a(n-3) - 36*a(n-4) + 24*a(n-5) - 8*a(n-6) for n > 5.
%F E.g.f.: (2 + exp(x)*((1 + x)*(2 + x)*cos(x) - (1 + x + x^2)*sin(x)))/4. (End)
%e a(3) = -16:
%e [ 1, 2*i, 3*i;
%e -2*i, 1, 2*i;
%e -3*i, -2*i, 1 ]
%t Join[{1},Table[Det[ToeplitzMatrix[Join[{1},I Range[2,n]]]],{n,36}]]
%o (PARI) a(n) = matdet(matrix(n, n, i, j, if (i==j, 1, if (i<j, I*(j-i+1), I*(j-i-1))))); \\ _Michel Marcus_, Jan 20 2023
%o (Python)
%o from sympy import Matrix, I
%o def A359559(n): return Matrix(n,n,[i-j+(1 if i>j else -1) if i!=j else I for i in range(n) for j in range(n)]).det()*(1,-I,-1,I)[n&3] # _Chai Wah Wu_, Jan 25 2023
%Y Cf. A001792 (symmetric Toeplitz matrix), A143182.
%Y Cf. A359560 (permanent), A359561, A359562.
%Y Cf. A359614 (minimal), A359615 (maximal).
%K sign
%O 0,3
%A _Stefano Spezia_, Jan 06 2023
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