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

%I A086459 #14 Feb 16 2025 08:32:50
%S A086459 1,-3,49,-3375,923521,-992436543,4195872914689,-70110209207109375,
%T A086459 4649081944211090042881,-1227102111503512992112190463,
%U A086459 1291749870339606615892191271170049,-5429914198235566686555216227881787109375
%N A086459 Determinant of the circulant matrix whose rows are formed by successively rotating the vector (1, 2, 4, 8, ..., 2^(n-1)) right.
%C A086459 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.
%D A086459 Richard Bellman, Introduction to Matrix Analysis, Second Edition, SIAM, 1970, pp. 242-3.
%D A086459 Philip J. Davis, Circulant Matrices, Second Edition, Chelsea, 1994.
%H A086459 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CirculantMatrix.html">Circulant Matrix</a>
%F A086459 a(n) = (-2^n + 1)^(n-1).
%F A086459 See formulas in A180602, an unsigned version of this sequence with offset 0. [_Paul D. Hanna_, Sep 11 2010]
%e A086459 a(3) = determinant of the matrix ((1,2,4),(4,1,2),(2,4,1)) = 49. [Corrected by _T. D. Noe_, Jan 22 2008]
%p A086459 restart:with (combinat):a:=n->mul(-stirling2(n,2), j=3..n): seq(a(n), n=2..19); # _Zerinvary Lajos_, Jan 01 2009
%t A086459 Table[x=2^Range[0, n-1]; m=Table[RotateRight[x, i-1], {i, n}]; Det[m], {n, 12}]
%Y A086459 Cf. A048954 (circulant of binomial coefficients), A052182 (circulant of natural numbers), A066933 (circulant of prime numbers).
%Y A086459 Cf. A180602 (unsigned, offset 0). [_Paul D. Hanna_, Sep 11 2010]
%K A086459 easy,sign
%O A086459 1,2
%A A086459 _T. D. Noe_, Jul 21 2003