%I #53 May 24 2022 11:00:33
%S 4,48,1440,80640,7257600,958003200,174356582400,41845579776000,
%T 12804747411456000,4865804016353280000,2248001455555215360000,
%U 1240896803466478878720000,806582922253211271168000000,609776689223427721003008000000,530505719624382117272616960000000,526261673867387060334436024320000000
%N Value of the permanent of the matrix [1-zeta^{j-k}]_{1<=j,k<=2n}, where zeta is any primitive 2n-th root of unity.
%C The author has proved that the exact value of a(n) is 2*(2n)!. Moreover, for any primitive n-th root zeta of unity, the permanent of the matrix [1-zeta^j*x_k]_{1<=j,k<=n} is n!(1-x_1..x_n).
%C Conjecture: Let zeta be a primitive 2n-th root of unity. Then the sum of those Product_{j=1..2n}(1-zeta^{j-f(j)})^{-1} with f over all the derangements of {1,...,2n} has the exact value ((2n-1)!!/2^n)^2.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/2108.07723">Arithmetic properties of some permanents</a>, arXiv:2108.07723 [math.GM], 2021.
%F a(n) = 2*(2*n)!.
%e a(1) is the permanent of the matrix [1-(-1)^{1-1},1-(-1)^{1-2};1-(-1)^{2-1},1-(-1)^{2-2}] = [0,2;2,0], which equals 4.
%t a[n_]:=a[n]= Permanent[Table[1-E^(2*Pi*I*(j-k)/(2*n)),{j,1,2n},{k,1,2n}]];
%t (* Though a(n) is actually an integer, Mathematica could not find its exact value for a general positive integer n. Instead, we may check approximate values of a(n) such as N[a[5],10] = 7257600.000. *)
%o (PARI) default(realprecision, 100); a(n) = round(real(matpermanent(matrix(2*n, 2*n, j, k, 1-exp(Pi*I*(j-k)/n))))) \\ _Michel Marcus_, Aug 08 2021
%Y Cf. A000166, A003112, A204249, A278847, A346934.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Aug 08 2021
%E a(16) from _Vaclav Kotesovec_, Aug 21 2021