0,5

In the Shevelev link the author proved that, for odd N>=3 and every n>=1, the determinant of circulant matrix of order N with entries in the first row (-1)^j*Sum{k>=0}(-1)^k*binomial(n, N*k+j), j=0..N-1, is 0.

This sequence shows what happens for the first even N>3.

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

a(n) = 0 for n == 3 (mod 4).

G.f. (empirical): (1/8)*(68*x^2+1)/(16*x^4+136*x^2+1)+(1/4)*(68*x^2-8*x+1)/(16*x^4+64*x^3+128*x^2-16*x+1)+(1/2)*(12*x^2+1)/(16*x^4+24*x^2+1)+3/(8*(4*x^2+1))-(1/4)*(12*x^2-4*x+1)/(16*x^4-32*x^3+32*x^2-8*x+1)-(1/4)*(4*x^2+1)/(16*x^4+1)+(1/4)*(12*x^2+4*x+1)/(16*x^4+32*x^3+32*x^2+8*x+1). - Robert Israel, Jul 26 2017

seq(LinearAlgebra:-Determinant(Matrix(4, shape=Circulant[seq((-1)^j*

add((-1)^k*binomial(n, 4*k+j), k=0..n/4), j=0..3)])), n=0..50); # Robert Israel, Jul 26 2017

ro[n_] := Table[Sum[(-1)^(j+k) Binomial[n, 4k+j], {k, 0, n/4}], {j, 0, 3}];

M[n_] := Table[RotateRight[ro[n], m], {m, 0, 3}];

a[n_] := Det[M[n]];

Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 09 2018 *)

(Python)

from sympy.matrices import Matrix

from sympy import binomial

def mj(j, n): return (-1)**j*sum((-1)**k*binomial(n, 4*k + j) for k in range(n//4 + 1))

def a(n):

m=Matrix(4, 4, lambda i, j: mj((i-j)%4, n))

return m.det()

print([a(n) for n in range(22)]) # Indranil Ghosh, Jul 31 2017

Cf. A099586 (prefixed by a(0)=1), A099587, A099588, A099589, A290285.

Sequence in context: A254973 A092924 A331770 * A187863 A280869 A145235

Adjacent sequences: A290283 A290284 A290285 * A290287 A290288 A290289

Vladimir Shevelev and Peter J. C. Moses, Jul 26 2017

approved