%I
%S 1,0,0,0,0,0,0,0,0,0,2276485387658524,523547340003805770400,
%T 39617190432735671861429500,2896792542975174202888623380000,
%U 95819032881785191861991031568287500,1018409199709889673458815786392849200000
%N Determinant of circulant matrix of order 10 with entries in the first row that are (1)^(j1)*Sum_{k>=0} (1)^k*binomial(n, 10*k+j1), for j=1..10.
%C a(n) = 0 for n == 9 (mod 10).
%C A generalization. For an even N >= 2, consider the determinant of circulant matrix of order N with entries in the first row (1)^(j1)K_j(n), j=1..N, where K_j(n) = Sum_{k>=0} (1)^k*binomial(n, N*k+j1). Then it is 0 for n == N1 (mod N). This statement follows from an easily proved identity K_j(N*t + N  1) = (1)^t*K_(N  j + 1)(N*t + N  1) and a known calculation formula for the determinant of circulant matrix [Wikipedia]. Besides, it is 0 for n=1..N2. We also conjecture that every such sequence contains infinitely many blocks of N1 negative and N1 positive terms separated by 0's.
%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 f:= n > LinearAlgebra:Determinant(Matrix(10,10,shape=
%p Circulant[seq((1)^j*add((1)^k*binomial(n,10*k+j),
%p k=0..(nj)/10), j=0..9)])):
%p map(f, [$0..20]); # _Robert Israel_, Aug 08 2017
%t ro[n_] := Table[(1)^(j1) Sum[(1)^k Binomial[n, 10k+j1], {k, 0, n/10}], {j, 1, 10}];
%t M[n_] := Table[RotateRight[ro[n], m], {m, 0, 9}];
%t a[n_] := Det[M[n]];
%t Table[a[n], {n, 0, 15}] (* _JeanFrançois Alcover_, Aug 10 2018 *)
%Y Cf. A290286, A290535, A290539.
%K sign
%O 0,11
%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Aug 05 2017
