A118709 a(n) = determinant of n X n circulant matrix whose first row is the first n cube numbers 0, 1, ..., (n-1)^3.

0, -1, 513, -532800, 1077540500, -3831689610000, 22051842087895137, -192710430555501494272, 2433436736207275231050384, -42684202683959414242500000000, 1007311823853329619224620155226025, -31149342348518897782279760206406615040

Offset: 1

Author

Jonathan Vos Post, May 20 2006

Keywords

• easy
• sign

Examples

			a(2) = -1 because of the determinant -1 =
| 0, 1 |
| 1, 0 |.
a(3) = 513 = determinant
|0,1,8|
|8,0,1|
|1,8,0|.
a(6) = 22051842087895137 = determinant
|0,1,8,27,64,125,216|
|216,0,1,8,27,64,125|
|125,216,0,1,8,27,64|
|64,125,216,0,1,8,27|
|27,64,125,216,0,1,8|
|8,27,64,125,216,0,1|
|1,8,27,64,125,216,0|.
		

Links

• Eric Weisstein's World of Mathematics, Circulant Matrix.

Crossrefs

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.

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

Programs

Mathematica

Table[Det[Table[RotateRight[Range[0,i]^3,n],{n,0,i}]],{i,0,10}] (* Harvey P. Dale, Oct 22 2012 *)

Formula

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

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

where

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

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

Extensions

More terms from Harvey P. Dale, Oct 22 2012