A290535 -id:A290535 - cp's OEIS frontend

A290539 Determinant of circulant matrix of order eight with entries in the first row that are (-1)^(j-1) * Sum_{k>=0} (-1)^kbinomial(n,8k+j-1), for j=1..8.

1, 0, 0, 0, 0, 0, 0, 0, -8489565952, -31872959692800, -932158289501356032, -4169183582652459909120, -5144394740685202662359040, -2505627397073121215653085184, -500556279165026162974748835840, 0, 20396260728315877590754520243175424

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Aug 05 2017

sign

a(n) = 0 for n == 7 (mod 8).

Robert Israel, Table of n, a(n) for n = 0..428
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Wikipedia, Circulant matrix

Cf. A290285, A290286, A290535, A290540.

seq(LinearAlgebra:-Determinant(Matrix(8,8,shape=Circulant[seq(
(-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

ro[n_] := Table[(-1)^(j-1) Sum[(-1)^k*Binomial[n, 8k+j-1], {k, 0, n/8}], {j, 1, 8}];
M[n_] := Table[RotateRight[ro[n], m], {m, 0, 7}];
a[n_] := Det[M[n]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 10 2018 *)

A290540 Determinant of circulant matrix of order 10 with entries in the first row that are (-1)^(j-1)Sum_{k>=0} (-1)^kbinomial(n, 10*k+j-1), for j=1..10.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2276485387658524, -523547340003805770400, -39617190432735671861429500, -2896792542975174202888623380000, -95819032881785191861991031568287500, -1018409199709889673458815786392849200000

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Aug 05 2017

sign

a(n) = 0 for n == 9 (mod 10).

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.

Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Wikipedia, Circulant matrix

Cf. A290286, A290535, A290539.

f:= n -> LinearAlgebra:-Determinant(Matrix(10,10,shape=
  Circulant[seq((-1)^j*add((-1)^k*binomial(n,10*k+j),
     k=0..(n-j)/10), j=0..9)])):
map(f, [$0..20]); # Robert Israel, Aug 08 2017

ro[n_] := Table[(-1)^(j-1) Sum[(-1)^k Binomial[n, 10k+j-1], {k, 0, n/10}], {j, 1, 10}];
M[n_] := Table[RotateRight[ro[n], m], {m, 0, 9}];
a[n_] := Det[M[n]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Aug 10 2018 *)

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A290539 Determinant of circulant matrix of order eight with entries in the first row that are (-1)^(j-1) * Sum_{k>=0} (-1)^kbinomial(n,8k+j-1), for j=1..8.

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A290540 Determinant of circulant matrix of order 10 with entries in the first row that are (-1)^(j-1)Sum_{k>=0} (-1)^kbinomial(n, 10*k+j-1), for j=1..10.

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cp's OEIS Frontend

A290539 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.

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Maple

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A290540 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.

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Mathematica

A290539 Determinant of circulant matrix of order eight with entries in the first row that are (-1)^(j-1) * Sum_{k>=0} (-1)^kbinomial(n,8k+j-1), for j=1..8.

A290540 Determinant of circulant matrix of order 10 with entries in the first row that are (-1)^(j-1)Sum_{k>=0} (-1)^kbinomial(n, 10*k+j-1), for j=1..10.