|
%I #32 Apr 23 2021 01:21:12
%S 1,0,0,62,666,5292,39754,307062,2456244,19825910,159305994,1274445900,
%T 10184391946,81430393590,651443132340,5212260963062,41700950994186,
%U 333607607822412,2668815050206474,21350337149539062,170802697195263924,1366424509598012150
%N 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.
%C 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.
%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 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
%p a:= n-> LinearAlgebra[Determinant](Matrix(3, shape=Circulant[seq(
%p (-1)^j*add(binomial(n, 3*k+j), k=0..(n-j)/3), j=0..2)])):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Jul 27 2017
%t ro[n_] := Table[(-1)^j Sum[Binomial[n, 3k+j], {k, 0, n/3}], {j, 0, 2}];
%t M[n_] := Table[RotateRight[ro[n], m], {m, 0, 2}];
%t a[n_] := Det[M[n]];
%t Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Aug 09 2018 *)
%o (PARI) mj(j,n) = (-1)^j*sum(k=0, n\3, binomial(n, 3*k+j));
%o 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
%o (Python)
%o from sympy.matrices import Matrix
%o from sympy import binomial
%o def mj(j, n):
%o return (-1)**j*sum(binomial(n, 3*k + j) for k in range(n//3 + 1))
%o def a(n):
%o m=Matrix(3, 3, [0]*9)
%o for j in range(3):m[0, j]=mj(j, n)
%o for j in range(1, 3):m[1, j]=m[0, j - 1]
%o m[1, 0]=m[0, 2]
%o for j in range(1, 3):m[2, j] = m[1, j - 1]
%o m[2, 0]=m[1, 2]
%o return m.det()
%o print([a(n) for n in range(22)]) # _Indranil Ghosh_, Jul 31 2017
%Y Cf. A024493, A024495, A131708, A290286.
%K nonn
%O 0,4
%A _Vladimir Shevelev_, Jul 26 2017
%E More terms from _Peter J. C. Moses_, Jul 26 2017
|
