Gradshteyn and Ryzhik define the circulant determinant by left bracketing bar x_1 | x_2 | x_3 | ... | x_n x_n | x_1 | x_2 | ... | x_(n - 1) x_(n - 1) | x_n | x_1 | ... | x_(n - 2) ⋮ | ⋮ | ⋮ | ⋱ | ⋮ x_2 | x_3 | x_4 | ... | x_1 right bracketing bar = product_(j = 1)(x_1 + x_2 ω_j + x_3 ω_j^2 + ... + x_n ω_j^(n - 1)), where ω_j is the nth root of unity.