Search: id:A086569
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%I A086569
%S A086569 1,3,28,375,3751,49392,6835648,1343091375,364668913756,210736858987743,
%T A086569 101832157445630503,67043511427995648000,487627751563388801409591,
%U A086569 4875797582053878382039400448,58623274842128064372315087290368
%V A086569 1,-3,28,-375,3751,-49392,6835648,-1343091375,364668913756,-210736858987743,
%W A086569 101832157445630503,-67043511427995648000,487627751563388801409591,
%X A086569 -4875797582053878382039400448,58623274842128064372315087290368
%N 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.
%C A086569 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 A086569 For references, see A086459
%e A086569 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 A086569 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 A086569 Cf. A048954, A086459 (circulant of powers of 2).
%Y A086569 Sequence in context: A151423 A161605 A048954 this_sequence A143636 A060545 
               A108288
%Y A086569 Adjacent sequences: A086566 A086567 A086568 this_sequence A086570 A086571 
               A086572
%K A086569 easy,sign
%O A086569 1,2
%A A086569 T. D. Noe (noe(AT)sspectra.com), Jul 21 2003

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