A15484. Doremy's Pegging Game

Doremy has $n+1$ pegs. There are $n$ red pegs arranged as vertices of a regular $n$ -sided polygon, numbered from $1$ to $n$ in anti-clockwise order. There is also a blue peg of slightly smaller diameter in the middle of the polygon. A rubber band is stretched around the red pegs.

Doremy is very bored today and has decided to play a game. Initially, she has an empty array $a$ . While the rubber band does not touch the blue peg, she will:

1. choose $i$ ( $1 \leq i \leq n$ ) such that the red peg $i$ has not been removed;
2. remove the red peg $i$ ;
3. append $i$ to the back of $a$ .

Doremy wonders how many possible different arrays $a$ can be produced by the following process. Since the answer can be big, you are only required to output it modulo $p$ . $p$ is guaranteed to be a prime number.

 game with $n=9$ and $a=[7,5,2,8,3,9,4]$ and another game with $n=8$ and $a=[3,4,7,1,8,5,2]$

The first line contains two integers $n$ and $p$ ( $3 \leq n \leq 5000$ , $10^8 \le p \le 10^9$ ) — the number of red pegs and the modulo respectively.

$p$ is guaranteed to be a prime number.

Output a single integer, the number of different arrays $a$ that can be produced by the process described above modulo $p$ .

In the first test case, $n=4$ , some possible arrays $a$ that can be produced are $[4,2,3]$ and $[1,4]$ . However, it is not possible for $a$ to be $[1]$ or $[1,4,3]$ .