Conjugation polynomials

[0, 0, 0, 0, 1, 1, 1, 1] 1/24*(x - 1)(x - 2)(x - 3)x [0, 0, 0, 1, 0, 0, 0, 1] 1/6(x - 1)(x - 2)x [0, 0, 0, 1, 1, 1, 1, 0] -1/24(3x - 13)(x - 1)(x - 2)x [0, 0, 1, 0, 0, 0, 1, 0] -1/2(x - 1)(x - 3)x [0, 0, 1, 0, 1, 1, 0, 1] 1/24(7x - 26)(x - 1)(x - 3)x [0, 0, 1, 1, 0, 0, 1, 1] 1/2(x - 1)x [0, 0, 1, 1, 1, 1, 0, 0] 1/24(3x^2 - 23x + 46)(x - 1)x [0, 1, 0, 0, 0, 1, 0, 0] 1/2(x - 2)(x - 3)x [0, 1, 0, 0, 1, 0, 1, 1] -1/8(x - 2)(x - 3)(x - 5)x [0, 1, 0, 1, 0, 1, 0, 1] x [0, 1, 0, 1, 1, 0, 1, 0] -1/24(7x^2 - 44x + 61)(x - 2)x [0, 1, 1, 0, 0, 1, 1, 0] -1/2(x - 3)x [0, 1, 1, 0, 1, 0, 0, 1] 1/8(x^2 - 3x - 2)(x - 3)x [0, 1, 1, 1, 0, 1, 1, 1] 1/6(x^2 - 6x + 11)x [0, 1, 1, 1, 1, 0, 0, 0] -1/24(x^2 - 5x + 10)(x - 5)*x