A290540 - OEIS

%I #36 Aug 10 2018 02:36:51

%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)^(j-1)*Sum_{k>=0} (-1)^k*binomial(n, 10*k+j-1), 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)^(j-1)K_j(n), j=1..N, where K_j(n) = Sum_{k>=0} (-1)^k*binomial(n, N*k+j-1). Then it is 0 for n == N-1 (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..N-2. We also conjecture that every such sequence contains infinitely many blocks of N-1 negative and N-1 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..(n-j)/10), j=0..9)])):

%p map(f, [$0..20]); # _Robert Israel_, Aug 08 2017

%t ro[n_] := Table[(-1)^(j-1) Sum[(-1)^k Binomial[n, 10k+j-1], {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}] (* _Jean-François 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