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%I #25 Aug 10 2018 02:36:37
%S 1,0,0,0,0,0,0,0,-8489565952,-31872959692800,-932158289501356032,
%T -4169183582652459909120,-5144394740685202662359040,
%U -2505627397073121215653085184,-500556279165026162974748835840,0,20396260728315877590754520243175424
%N Determinant of circulant matrix of order eight with entries in the first row that are (-1)^(j-1) * Sum_{k>=0} (-1)^k*binomial(n,8*k+j-1), for j=1..8.
%C a(n) = 0 for n == 7 (mod 8).
%H Robert Israel, <a href="/A290539/b290539.txt">Table of n, a(n) for n = 0..428</a>
%H Vladimir Shevelev, <a href="https://arxiv.org/abs/1706.01454">Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n</a>, arXiv:1706.01454 [math.CO], 2017.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Circulant_matrix">Circulant matrix</a>
%p seq(LinearAlgebra:-Determinant(Matrix(8,8,shape=Circulant[seq(
%p (-1)^(j-1)*add((-1)^k*binomial(n,8*k+j-1),k=0..n/8),j=1..8)])), n=0..20); # _Robert Israel_, Aug 11 2017
%t ro[n_] := Table[(-1)^(j-1) Sum[(-1)^k*Binomial[n, 8k+j-1], {k, 0, n/8}], {j, 1, 8}];
%t M[n_] := Table[RotateRight[ro[n], m], {m, 0, 7}];
%t a[n_] := Det[M[n]];
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Aug 10 2018 *)
%Y Cf. A290285, A290286, A290535, A290540.
%K sign
%O 0,9
%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Aug 05 2017
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