A346949 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.

4, 48, 1440, 80640, 7257600, 958003200, 174356582400, 41845579776000, 12804747411456000, 4865804016353280000, 2248001455555215360000, 1240896803466478878720000, 806582922253211271168000000, 609776689223427721003008000000, 530505719624382117272616960000000, 526261673867387060334436024320000000

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

Table of n, a(n) for n=1..16.

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

Cf. A000166, A003112, A204249, A278847, A346934.

Sequence in context: A013150 A353922 A332865 * A011266 A360820 A362828

Adjacent sequences: A346946 A346947 A346948 * A346950 A346951 A346952

Keyword

nonn

Author

Zhi-Wei Sun, Aug 08 2021

Extensions

a(16) from Vaclav Kotesovec, Aug 21 2021

Status

approved