

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.


8



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

1,2


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 nspace. However, the volume in n2 space is positive and is given by the product of the nonzero eigenvalues.


REFERENCES

For references, see A086459


LINKS

Table of n, a(n) for n=1..15.


EXAMPLE

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}}.


MATHEMATICA

Table[x=Binomial[n, Range[0, n1]]; m=Table[RotateRight[x, i1], {i, n}]; e=Eigenvalues[m]; prod=1; Do[If[e[[i]]!=0, prod=prod*e[[i]]], {i, n}]; FullSimplify[prod], {n, 15}]


CROSSREFS

Cf. A048954, A086459 (circulant of powers of 2).
Sequence in context: A151423 A161605 A048954 * A264639 A298696 A143636
Adjacent sequences: A086566 A086567 A086568 * A086570 A086571 A086572


KEYWORD

easy,sign


AUTHOR

T. D. Noe, Jul 21 2003


STATUS

approved



