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A086459
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Determinant of the circulant matrix whose rows are formed by successively rotating the vector (1,2,4,8,..2^(n-1)) right.
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10
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1, -3, 49, -3375, 923521, -992436543, 4195872914689, -70110209207109375, 4649081944211090042881, -1227102111503512992112190463, 1291749870339606615892191271170049, -5429914198235566686555216227881787109375
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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.
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REFERENCES
| Richard Bellman, Introduction to Matrix Analysis, Second Edition, SIAM, 1970, pp. 242-3.
Philip J. Davis, Circulant Matrices, Second Edition, Chelsea, 1994.
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LINKS
| Eric Weisstein's World of Mathematics, Circulant Matrix
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FORMULA
| a(n) = (-2^n+1)^(n-1)
See formulas in A180602, an unsigned version of this sequence with offset 0. [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 11 2010]
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EXAMPLE
| a(3) = determinant of the matrix ((1,2,4),(4,1,2),(2,4,1)) = 49. [Corrected by T. D. Noe (noe(AT)sspectra.com), Jan 22 2008]
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MAPLE
| restart:with (combinat):a:=n->mul(-stirling2(n, 2), j=3..n): seq(a(n), n=2..19); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 01 2009]
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MATHEMATICA
| Table[x=2^Range[0, n-1]; m=Table[RotateRight[x, i-1], {i, n}]; Det[m], {n, 12}]
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CROSSREFS
| Cf. A048954 (circulant of binomial coefficients), A052182 (circulant of natural numbers), A066933 (circulant of prime numbers).
Cf. A180602 (unsigned, offset 0). [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 11 2010]
Sequence in context: A106842 A203743 A180602 * A203700 A063893 A145572
Adjacent sequences: A086456 A086457 A086458 * A086460 A086461 A086462
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KEYWORD
| easy,sign
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Jul 21 2003
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