In a previous MO question, the OP asks a proof for $\det_{1\leq i,j\leq n}\left(\binom{i}{2j}+\binom{-i}{2j}\right)=1$. Subsequently, Gjergji Zaimi generalized the problem to $$\det_{1\le i,j\le n}\left( \binom{x_i}{2j}+ \binom{-x_i}{2j}\right)=\prod_{i=1}^n x_i^2 \prod_{i<j} (x_j^2-x_i^2) \prod_{j=1}^n \frac{2}{(2j)!}.$$ Let's denote $y_k:=x_k^2$ and also $$\Psi_n:=\prod_{i=1}^n y_i \prod_{i<j} (y_j-y_i) \prod_{j=1}^{\mathbf{n+1}} \frac{2}{(2j)!}.$$ Recall the elementary symmetric functions $e_k=e_k(y_1,\dots,y_n)$. Now, I would like ask:

Question 1. Is it true that there exist positive integers $a_0, a_1, \dots, a_n$ such that $$\det_{1\le i,j\le n}\left( \binom{x_i}{2j+2}+ \binom{-x_i}{2j+2}\right)= (a_0+a_1e_1+\cdots+a_{n-1}e_{n-1}+a_ne_n)\cdot\Psi_n.$$ Question 2. What are these coefficients $a_0, a_1, \dots, a_n$?

NOTE. For any $n$, we observe that $a_n=1, a_{n-1}=11, a_{n-2}=661, a_{n-3}=151451$. These sequence does not appear on OEIS.

In a previous MO question, the OP asks a proof for $\det_{1\leq i,j\leq n}\left(\binom{i}{2j}+\binom{-i}{2j}\right)=1$. Subsequently, Gjergji Zaimi generalized the problem to $$\det_{1\le i,j\le n}\left( \binom{x_i}{2j}+ \binom{-x_i}{2j}\right)=\prod_{i=1}^n x_i^2 \prod_{i<j} (x_j^2-x_i^2) \prod_{j=1}^n \frac{2}{(2j)!}.$$ Let's define $$\Psi_n:=\prod_{i=1}^n x_i^2 \prod_{i<j} (x_j^2-x_i^2) \prod_{j=1}^{\mathbf{n+1}} \frac{2}{(2j)!}.$$ Recall the elementary symmetric functions $e_k=e_k(x_1^2,\dots,x_n^2)$. Now, I would like ask:

Question 1. Is it true that there exist positive integers $a_0, a_1, \dots, a_n$ such that $$\det_{1\le i,j\le n}\left( \binom{x_i}{2j+2}+ \binom{-x_i}{2j+2}\right)= (a_0+a_1e_1+\cdots+a_{n-1}e_{n-1}+a_ne_n)\cdot\Psi_n.$$ Question 2. What are these coefficients $a_0, a_1, \dots, a_n$?

NOTE. For any $n$, we observe that $a_n=1, a_{n-1}=11, a_{n-2}=661, a_{n-3}=151451$. These sequence does not appear on OEIS.