A322898 a(n) is the permanent of the matrix [((i + j)/(2*n + 1))]_{i,j=0..n}, where (k/m) denotes the Jacobi symbol.

1, 1, 2, 2, 20, 16, 48, 55, 128, 320, 1206, 768, 406446336, 43545600, 141312, 2267136, 389112, 1624232, 138739712, 122605392, 2262695936, 20313407488, 17060393728, 189261676544, 374345132371011500507136, 669835780976, -7000008163328, 22712032822272, 2244036651776, 4363027965018112, 30229121955004416, -46693326700068864, -23328907207088128, 3173005987716005888, 136427303851761536

Offset

0,3

Comments

Conjecture: a(n) is positive for any nonnegative integer n.

This conjecture is false, a(26) < 0. - Vaclav Kotesovec, Aug 20 2021

Links

Table of n, a(n) for n=0..34.

Zhi-Wei Sun, Is the permanent of the matrix [((i+j)/(2n+1))]_{0<=i,j<=n} always positive?, Question 319745, December 30, 2018.

Example

a(1) = 1 since the entries of the matrix A = [Jacobi(i+j,2*1+1)]_{i,j=0,1} are 0, 1 (in the first row) and 1, -1 (in the second row), and per(A) = 0*(-1) + 1*1 = 1.

Mathematica

Permanent[m_List]:=With[{v = Array[x, Length[m]]}, Coefficient[Times @@ (m.v), Times @@ v]];
a[n_]:=a[n]=Permanent[Table[JacobiSymbol[i+j, 2n+1], {i, 0, n}, {j, 0, n}]];
Do[Print[n, " ", a[n]], {n, 0, 25}]

Prog

(PARI) a(n) = matpermanent(matrix(n+1, n+1, i, j, i--; j--; kronecker(i+j, 2*n+1))) \\ Michel Marcus, Dec 30 2018

Crossrefs

Cf. A322363.

Sequence in context: A326177 A370709 A103129 * A009340 A053593 A002907

Adjacent sequences: A322895 A322896 A322897 * A322899 A322900 A322901

Keyword

sign

Author

Zhi-Wei Sun, Dec 30 2018

Extensions

a(26)-a(34) from Vaclav Kotesovec, Aug 20 2021

Status

approved