%I #34 Aug 27 2022 08:13:04

%S 1,-10,5870,-436619903,204409938157631,-445211270783816444430983,

%T 396293055917701185511708646165266818589,

%U -50075241686736491417389691481681057226117081701278447,5775845962974711619751142332219720943292883679628990682130802753216873

%N Numerator of the permanent of the matrix [m(j,k)]_{j,k=0..2*n}, where m(j,k) is 1 or (j+k)/(j-k) according as j = k or not.

%C Conjecture 1. If n is a positive odd integer, x(0) = 0, and x(1),...,x(n) are variables, then the permanent of the matrix [x(j,k)]_{j,k=0..n} vanishes, where x(j,k) is 1 or (x(j)+x(k))/(x(j)-x(k)) according as j = k or not.

%C Conjecture 2: (-1)^n*a(n) > 0 for all n = 0,1,2,...

%C Conjecture 3: For any odd prime p, the permanent of the matrix [m(j,k)]_{j,k=0..p-1} is congruent to 1 - (-1)^((p-1)/2) modulo p, where m(j,k) is defined as in the title..

%C Conjecture 4: For any odd prime p, the permanent of the matrix M = [m(j,k)]_{j,k=1..p-1} is congruent to ((p-2)!!)^2 modulo p^2, and det(M) is congruent to ((p-2)!!)^2*(-1)^((p+1)/2)/(p-2) modulo p^2.

%H Yue-Feng She, Zhi-Wei Sun, and Wei Xia, <a href="https://arxiv.org/abs/2208.12167">A novel permanent identity</a>, arXiv:2208.12167 [math.CO], 2022.

%H Zhi-Wei Sun, <a href="http://hitpress.hit.edu.cn/2021/1015/c12593a261001/page.htm">New Conjectures in Number Theory and Combinatorics</a> (in Chinese), Harbin Institute of Technology Press, 2021.

%H Zhi-Wei Sun, <a href="https://arxiv.org/abs/2108.07723">Arithmetic properties of some permanents</a>, arXiv:2108.07723 [math.GM], 2021.

%e a(1) = -10 since the permanent of the matrix [m(j,k)]_{j,k=0,1,2} = [1,-1,1; 1,1,-3; 1,3,1] is -10.

%t a[n_]:=a[n]=Numerator[Permanent[Table[If[j==k,1,(j+k)/(j-k)],{j,0,2n},{k,0,2n}]]]

%t Table[a[n],{n,0,8}]

%o (PARI) a(n) = numerator(matpermanent(matrix(2*n+1, 2*n+1, i, j, i--; j--; if (i==j, 1, (i+j)/(i-j))))); \\ _Michel Marcus_, Jul 24 2022

%Y Cf. A204249.

%K sign

%O 0,2

%A _Zhi-Wei Sun_, Jul 24 2022