A086569 Product of the nonzero eigenvalues of the circulant matrix whose rows are formed by successively rotating a vector of binomial coefficients right. Generalization of A048954.

1, -3, 28, -375, 3751, -49392, 6835648, -1343091375, 364668913756, -210736858987743, 101832157445630503, -67043511427995648000, 487627751563388801409591, -4875797582053878382039400448, 58623274842128064372315087290368

Offset: 1

Author

T. D. Noe, Jul 21 2003

Keywords

• easy
• sign

Comments

In sequence A048954, a determinant of a circulant matrix, a(n) = 0 when 6 divides n. The determinant of a matrix can be interpreted as the signed volume of a simplex whose vertices are given by the rows of the matrix. For n a multiple of 6, the points form a lower dimensional simplex that has zero volume in n-space. However, the volume in n-2 space is positive and is given by the product of the nonzero eigenvalues.

Examples

			a(6) = -49392 because -1, -28, -28 and 63 are the four nonzero eigenvalues of the matrix {{1,6,15,20,15,6}, {6,1,6,15,20,15}, {15,6,1,6,15,20}, {20,15,6,1,6,15}, {15,20,15,6,1,6}, {6,15,20,15,6,1}}.
		

References

• For references, see A086459

Crossrefs

Cf. A048954, A086459 (circulant of powers of 2).

Programs

Mathematica

Table[x=Binomial[n, Range[0, n-1]]; m=Table[RotateRight[x, i-1], {i, n}]; e=Eigenvalues[m]; prod=1; Do[If[e[[i]]!=0, prod=prod*e[[i]]], {i, n}]; FullSimplify[prod], {n, 15}]