A14881. Marin and Anti-coprime Permutation

Marin wants you to count number of permutations that are beautiful. A beautiful permutation of length $n$ is a permutation that has the following property: $$$$ \gcd (1 \cdot p_1, \, 2 \cdot p_2, \, \dots, \, n \cdot p_n) > 1, $$ where $\\gcd$ is the <a href="https://en.wikipedia.org/wiki/Greatest_common_divisor">greatest common divisor</a>.</p><p>A permutation is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $\[2,3,1,5,4\]$ is a permutation, but $\[1,2,2\]$ is not a permutation ( $2$ appears twice in the array) and $\[1,3, 4\]$ is also not a permutation ( $n=3$ but there is $4$$$ in the array).

The first line contains one integer $t$ ( $1 \le t \le 10^3$ ) — the number of test cases.

Each test case consists of one line containing one integer $n$ ( $1 \le n \le 10^3$ ).

For each test case, print one integer — number of beautiful permutations. Because the answer can be very big, please print the answer modulo $998\,244\,353$ .

In first test case, we only have one permutation which is $[1]$ but it is not beautiful because $\gcd(1 \cdot 1) = 1$ .

In second test case, we only have one beautiful permutation which is $[2, 1]$ because $\gcd(1 \cdot 2, 2 \cdot 1) = 2$ .