A322898 - OEIS

%I #21 Aug 20 2021 02:56:23

%S 1,1,2,2,20,16,48,55,128,320,1206,768,406446336,43545600,141312,

%T 2267136,389112,1624232,138739712,122605392,2262695936,20313407488,

%U 17060393728,189261676544,374345132371011500507136,669835780976,-7000008163328,22712032822272,2244036651776,4363027965018112,30229121955004416,-46693326700068864,-23328907207088128,3173005987716005888,136427303851761536

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

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

%C This conjecture is false, a(26) < 0. - _Vaclav Kotesovec_, Aug 20 2021

%H Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/319745">Is the permanent of the matrix [((i+j)/(2n+1))]_{0<=i,j<=n} always positive?</a>, Question 319745, December 30, 2018.

%e 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.

%t Permanent[m_List]:=With[{v = Array[x, Length[m]]},Coefficient[Times @@ (m.v), Times @@ v]];

%t a[n_]:=a[n]=Permanent[Table[JacobiSymbol[i+j,2n+1],{i,0,n},{j,0,n}]];

%t Do[Print[n," ",a[n]],{n,0,25}]

%o (PARI) a(n) = matpermanent(matrix(n+1, n+1, i, j, i--; j--; kronecker(i+j, 2*n+1))) \\ _Michel Marcus_, Dec 30 2018

%Y Cf. A322363.

%K sign

%O 0,3

%A _Zhi-Wei Sun_, Dec 30 2018

%E a(26)-a(34) from _Vaclav Kotesovec_, Aug 20 2021