On $\frac{(-1)^{(n-1)/2}}n\mathrm{per}\left[\tan\pi\frac{j+k}n\right]_{1\le j,k\le n-1}$ with $n\in\{3,5,7,\ldots\}$

Question

Recall that the permanent of a matrix $A=[a_{j,k}]_{1\le j,k\le n}$ is given by $$\mathrm{per}(A)=\sum_{\tau\in S_n}\prod_{j=1}^na_{j,\tau(j)}.$$

Let $n$ be an odd integer greater than one. In 2019 I studied $$t(n):=\frac{(-1)^{(n-1)/2}}n\mathrm{per}\left[\tan\pi\frac{j+k}n\right]_{1\le j,k\le n-1}.$$ I have proved that $t(n)\in\mathbb Q$, and that $$t(p)\in\mathbb Z\ \ \ \text{and}\ \ \ t(p)\equiv-2\pmod p$$ for every odd prime $p$.

Question. Whether for any odd integer $n>1$ the number $t(n)$ is always a positive integer congruent to $1$ modulo $4$ ?

My numerical computation indicates that $$t(3)=1,\ t(5)=13,\ t(7)=285,\ t(9)=16569, \ t(11)=1218105,\ t(13)=164741445.$$ I guess the question has a positive answer.

Your comments are welcome!

Answer 1

This is not an answer.

For any odd integer $n>1$, let

$$f(n)=\frac{2(n!!)^2}{n(n+1)}\sum_{k=0}^{\frac{n-1}{2}}\frac{(-1)^k}{2k+1}.$$

Numerical computation indicates that

$$t(n)=f(n)$$

for $3 \leq n \leq 33.$

                  [t(3) = 1, f(3) = 1]

                 [t(5) = 13, f(5) = 13]

                [t(7) = 285, f(7) = 285]

              [t(9) = 16569, f(9) = 16569]

           [t(11) = 1218105, f(11) = 1218105]

         [t(13) = 164741445, f(13) = 164741445]

       [t(15) = 25826325525, f(15) = 25826325525]

     [t(17) = 6310503824625, f(17) = 6310503824625]

  [t(19) = 1715718264685425, f(19) = 1715718264685425]

[t(21) = 661307821613200125, f(21) = 661307821613200125]

           [t(23) = 277040409578593786125,

            f(23) = 277040409578593786125]

          [t(25) = 154737143033349764435625, 

            f(25) = 154737143033349764435625]


         [t(27) = 92522564948525932110515625, 

           f(27) = 92522564948525932110515625]


        [t(29) = 70653112977123214268408803125, 

          f(29) = 70653112977123214268408803125]


      [t(31) = 57152351281142850164344240453125, 

        f(31) = 57152351281142850164344240453125]


     [t(33) = 57193813473884499796503057704390625, 

       f(33) = 57193813473884499796503057704390625]

Conj. For any odd integer $n>1$, $t(n)=f(n)$.