A290286 - OEIS

%I #30 Apr 12 2021 08:32:15

%S 1,0,0,0,-1008,-37120,-473600,0,63996160,702013440,2893578240,0,

%T -393379835904,-12971004067840,-160377313820672,0,21792325059543040,

%U 239501351489372160,987061897553510400,0,-134124249770961666048,-4422152303189489090560

%N Determinant of circulant matrix of order 4 with entries in the first row (-1)^j*Sum_{k>=0}(-1)^k*binomial(n, 4*k+j), j=0,1,2,3.

%C 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.

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

%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>

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

%F 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

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

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

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

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

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

%t Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Aug 09 2018 *)

%o (Python)

%o from sympy.matrices import Matrix

%o from sympy import binomial

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

%o def a(n):

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

%o     return m.det()

%o print([a(n) for n in range(22)]) # _Indranil Ghosh_, Jul 31 2017

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

%K sign

%O 0,5

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Jul 26 2017