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a(n) = determinant of n X n circulant matrix whose first row is the first n cube numbers 0, 1, ..., (n-1)^3.
1

%I #9 Oct 22 2012 16:30:59

%S 0,-1,513,-532800,1077540500,-3831689610000,22051842087895137,

%T -192710430555501494272,2433436736207275231050384,

%U -42684202683959414242500000000,1007311823853329619224620155226025,-31149342348518897782279760206406615040

%N a(n) = determinant of n X n circulant matrix whose first row is the first n cube numbers 0, 1, ..., (n-1)^3.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CirculantMatrix.html">Circulant Matrix</a>.

%F Contribution from Missouri State University Problem-Solving Group (MSUPSG(AT)MissouriState.edu), May 05 2010: (Start)

%F a(n) = (-1)^(n-1)*(n-1)^2*n^(n-2)*(n^(2n)-b(n)^n-c(n)^n+(n^2-3n+3)^n)/24

%F where

%F b(n)=(2*n^2-3*n-3+sqrt(15n^2-18n-9)i)/2 and

%F c(n)=(2*n^2-3*n-3-sqrt(15n^2-18n-9)i)/2 (End)

%e a(2) = -1 because of the determinant -1 =

%e | 0, 1 |

%e | 1, 0 |.

%e a(3) = 513 = determinant

%e |0,1,8|

%e |8,0,1|

%e |1,8,0|.

%e a(6) = 22051842087895137 = determinant

%e |0,1,8,27,64,125,216|

%e |216,0,1,8,27,64,125|

%e |125,216,0,1,8,27,64|

%e |64,125,216,0,1,8,27|

%e |27,64,125,216,0,1,8|

%e |8,27,64,125,216,0,1|

%e |1,8,27,64,125,216,0|.

%t Table[Det[Table[RotateRight[Range[0,i]^3,n],{n,0,i}]],{i,0,10}] (* _Harvey P. Dale_, Oct 22 2012 *)

%Y See also: A000578 The cubes: a(n) = n^3. A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A066933 Circulant of prime numbers. A086459 Circulant of powers of 2.

%Y Cf. A000578, A048954, A052182, A066933, A086459, A086569.

%K easy,sign

%O 1,3

%A _Jonathan Vos Post_, May 20 2006

%E More terms from _Harvey P. Dale_, Oct 22 2012