A290286 -id:A290286 - cp's OEIS frontend

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

Vladimir Shevelev, Jul 26 2017

nonn

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.

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. A024493, A024495, A131708, A290286.

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

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 *)

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

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

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

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

A290535 Determinant of circulant matrix of order 6 with entries in the first row (-1)^(j-1)Sum_{k>=0} (-1)^kbinomial(n, 6*k + j - 1), j=1..6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, -104284, -783050688, -329029322076, -43271152876224, -2175830808446736, 0, 5427970251634650916, 307609249050423946080, 8866068073884849492756, 137518739026000524646272, 896278292839676023110288, 0, -2518571790589921864549097500

Offset: 0

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

sign

a(n) = 0 for n == 5 (mod 6).

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.

ro[n_] := Table[(-1)^(j-1) Sum[(-1)^k Binomial[n, 6k+j-1], {k, 0, n/6}], {j, 1, 6}];
M[n_] := Table[RotateRight[ro[n], m], {m, 0, 5}];
a[n_] := Det[M[n]];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Aug 10 2018 *)

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.

Original entry on oeis.org

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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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.

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A290535 Determinant of circulant matrix of order 6 with entries in the first row (-1)^(j-1)*Sum_{k>=0} (-1)^k*binomial(n, 6*k + j - 1), j=1..6.

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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)^k*binomial(n,8*k+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)^k*binomial(n, 10*k+j-1), for j=1..10.

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A290535 Determinant of circulant matrix of order 6 with entries in the first row (-1)^(j-1)Sum_{k>=0} (-1)^kbinomial(n, 6*k + j - 1), j=1..6.

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.