|
Search: id:A086569
|
|
|
| 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. |
|
+0
9
|
|
| 1, -3, 28, -375, 3751, -49392, 6835648, -1343091375, 364668913756, -210736858987743, 101832157445630503, -67043511427995648000, 487627751563388801409591, -4875797582053878382039400448, 58623274842128064372315087290368
(list; graph; listen)
|
|
|
OFFSET
|
1,2
|
|
|
COMMENT
|
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.
|
|
REFERENCES
|
For references, see A086459
|
|
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, 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}]
|
|
CROSSREFS
|
Cf. A048954, A086459 (circulant of powers of 2).
Sequence in context: A151423 A161605 A048954 this_sequence A143636 A060545 A108288
Adjacent sequences: A086566 A086567 A086568 this_sequence A086570 A086571 A086572
|
|
KEYWORD
|
easy,sign
|
|
AUTHOR
|
T. D. Noe (noe(AT)sspectra.com), Jul 21 2003
|
|
|
Search completed in 0.002 seconds
|
