A290286 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.

1, 0, 0, 0, -1008, -37120, -473600, 0, 63996160, 702013440, 2893578240, 0, -393379835904, -12971004067840, -160377313820672, 0, 21792325059543040, 239501351489372160, 987061897553510400, 0, -134124249770961666048, -4422152303189489090560

Offset: 0

Author

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

Keywords

• sign

Comments

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.

Links

• 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

Crossrefs

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

Programs

Maple

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

Mathematica

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

Formula

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