A086459 Determinant of the circulant matrix whose rows are formed by successively rotating the vector (1, 2, 4, 8, ..., 2^(n-1)) right.

1, -3, 49, -3375, 923521, -992436543, 4195872914689, -70110209207109375, 4649081944211090042881, -1227102111503512992112190463, 1291749870339606615892191271170049, -5429914198235566686555216227881787109375

Offset: 1

Author

T. D. Noe, Jul 21 2003

Keywords

• easy
• sign

Comments

Note that if the rows are rotated left instead of right, the sign of the terms for which n = 0 or 3 (mod 4) is reversed. The n eigenvalues of these circulant matrices lie on the circle of radius 2(2^n - 1)/3 centered at x = (2^n - 1)/3, y = 0. This sequence can be generalized to bases other than 2 and similar results are true.

Examples

			a(3) = determinant of the matrix ((1,2,4),(4,1,2),(2,4,1)) = 49. [Corrected by _T. D. Noe_, Jan 22 2008]
		

References

• Richard Bellman, Introduction to Matrix Analysis, Second Edition, SIAM, 1970, pp. 242-3.
• Philip J. Davis, Circulant Matrices, Second Edition, Chelsea, 1994.

Links

• Eric Weisstein's World of Mathematics, Circulant Matrix

Crossrefs

Cf. A048954 (circulant of binomial coefficients), A052182 (circulant of natural numbers), A066933 (circulant of prime numbers).

Cf. A180602 (unsigned, offset 0). [Paul D. Hanna, Sep 11 2010]

Programs

Maple

restart:with (combinat):a:=n->mul(-stirling2(n,2), j=3..n): seq(a(n), n=2..19); # Zerinvary Lajos, Jan 01 2009

Mathematica

Table[x=2^Range[0, n-1]; m=Table[RotateRight[x, i-1], {i, n}]; Det[m], {n, 12}]

Formula

a(n) = (-2^n + 1)^(n-1).

See formulas in A180602, an unsigned version of this sequence with offset 0. [Paul D. Hanna, Sep 11 2010]