A290285 Determinant of circulant matrix of order 3 with entries in the first row (-1)^j * Sum_{k>=0} binomial(n,3*k+j), j=0,1,2.

1, 0, 0, 62, 666, 5292, 39754, 307062, 2456244, 19825910, 159305994, 1274445900, 10184391946, 81430393590, 651443132340, 5212260963062, 41700950994186, 333607607822412, 2668815050206474, 21350337149539062, 170802697195263924, 1366424509598012150

Offset: 0

Author

Vladimir Shevelev, Jul 26 2017

Keywords

• nonn

Comments

In the Shevelev link the author proved that, for even N>=2 and every n>=1, the determinant of circulant matrix of order N with entries in the first row being (-1)^j*Sum_{k>=0} binomial(n,N*k+j), j=0..N-1, is 0. This sequence shows what happens for the first odd N>2.

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. A024493, A024495, A131708, A290286.

Programs

Maple

a:= n-> LinearAlgebra[Determinant](Matrix(3, shape=Circulant[seq(
        (-1)^j*add(binomial(n, 3*k+j), k=0..(n-j)/3), j=0..2)])):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 27 2017

Mathematica

ro[n_] := Table[(-1)^j Sum[Binomial[n, 3k+j], {k, 0, n/3}], {j, 0, 2}];
M[n_] := Table[RotateRight[ro[n], m], {m, 0, 2}];
a[n_] := Det[M[n]];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 09 2018 *)

PARI

mj(j,n) = (-1)^j*sum(k=0, n\3, binomial(n, 3*k+j));
a(n) = {m = matrix(3, 3); for (j=1, 3, m[1, j] = mj(j-1,n)); for (j=2, 3, m[2, j] = m[1, j-1]); m[2, 1] = m[1, 3]; for (j=2, 3, m[3, j] = m[2, j-1]); m[3, 1] = m[2, 3]; matdet(m);} \\ Michel Marcus, Jul 26 2017

Python

from sympy.matrices import Matrix
from sympy import binomial
def mj(j, n):
    return (-1)**j*sum(binomial(n, 3*k + j) for k in range(n//3 + 1))
def a(n):
    m=Matrix(3, 3, [0]*9)
    for j in range(3):m[0, j]=mj(j, n)
    for j in range(1, 3):m[1, j]=m[0, j - 1]
    m[1, 0]=m[0, 2]
    for j in range(1, 3):m[2, j] = m[1, j - 1]
    m[2, 0]=m[1, 2]
    return m.det()
print([a(n) for n in range(22)]) # Indranil Ghosh, Jul 31 2017

Formula

G.f.: (1-12*x+48*x^2-73*x^3+6*x^4-60*x^5+736*x^6-576*x^7)/((1+x)*(-1+2*x)*(-1+8*x)* (1-x+x^2)*(1+2*x+4*x^2)*(1-4*x+16*x^2)). - Peter J. C. Moses, Jul 26 2017

Extensions

More terms from Peter J. C. Moses, Jul 26 2017