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%I #5 Mar 30 2012 17:22:28
%S 1,-3,28,-375,3751,-49392,6835648,-1343091375,364668913756,
%T -210736858987743,101832157445630503,-67043511427995648000,
%U 487627751563388801409591,-4875797582053878382039400448,58623274842128064372315087290368
%N 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.
%C 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.
%D For references, see A086459
%e 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}}.
%t 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}]
%Y Cf. A048954, A086459 (circulant of powers of 2).
%K easy,sign
%O 1,2
%A _T. D. Noe_, Jul 21 2003
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