OFFSET
1,1
COMMENTS
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).
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.
LINKS
Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.
FORMULA
a(n) = 2*(2*n)!.
EXAMPLE
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.
MATHEMATICA
a[n_]:=a[n]= Permanent[Table[1-E^(2*Pi*I*(j-k)/(2*n)), {j, 1, 2n}, {k, 1, 2n}]];
(* 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. *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 08 2021
EXTENSIONS
a(16) from Vaclav Kotesovec, Aug 21 2021
STATUS
approved