Xomu - Last Dance

⠀

You are given an integer $$$n$$$.

For each $$$(m, k)$$$ such that $$$3 \leq m \leq n+1$$$ and $$$0 \leq k \leq n-1$$$, count the permutations of $$$[1, 2, ..., n]$$$ such that $$$p_i + p_{i+1} \geq m$$$ for exactly $$$k$$$ indices $$$i$$$, modulo $$$998\,244\,353$$$.

Input

The input consists of a single line, which contains two integers $$$n$$$, $$$x$$$ ($$$2 \leq n \leq 4000$$$, $$$1 \leq x \lt 1\,000\,000\,007$$$).

Output

Let $$$a_{m,k}$$$ be the answer for the pair $$$(m, k)$$$, modulo $$$998\,244\,353$$$.

Let $$$$$$\large S = \sum_{m=3}^{n+1} \sum_{k=0}^{n-1} a_{m,k}x^{mn+k}\phantom{0}.$$$$$$

Output a single line with an integer: $$$S$$$ modulo $$$1\,000\,000\,007$$$.

Note that using two different modulos is intentional. We want you to calculate all the $$$a_{m,k}$$$ modulo $$$998\,244\,353$$$, then treat them like integers in the range $$$[0, 998\,244\,352]$$$, and hash them modulo $$$1\,000\,000\,007$$$.

Note

In the first test case, the answers for all $$$(m, k)$$$ are shown in the following table:

	$$$k = 0$$$	$$$k = 1$$$	$$$k = 2$$$
$$$m = 3$$$	$$$0$$$	$$$0$$$	$$$6$$$
$$$m = 4$$$	$$$0$$$	$$$4$$$	$$$2$$$

The answer for $$$(m, k) = (3, 2)$$$ is $$$6$$$, because for every permutation of length $$$3$$$, $$$a_i + a_{i+1} \geq 3$$$ exactly $$$2$$$ times.
The answer for $$$(m, k) = (4, 2)$$$ is $$$2$$$. In fact, there are $$$2$$$ permutations of length $$$3$$$ such that $$$a_i + a_{i+1} \geq 4$$$ exactly $$$2$$$ times: $$$[1, 3, 2]$$$, $$$[2, 3, 1]$$$.

Therefore, the value to print is $$$2^9 \cdot 0 + 2^{10} \cdot 0 + 2^{11} \cdot 6 + 2^{12} \cdot 0 + 2^{13} \cdot 4 + 2^{14} \cdot 2 \equiv 77\,824 \phantom{0} (\text{mod} \phantom{0} 1\,000\,000\,007)$$$.

In the second test case, the answers for all $$$(m, k)$$$ are shown in the following table:

	$$$k = 0$$$	$$$k = 1$$$	$$$k = 2$$$	$$$k = 3$$$
$$$m = 3$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$24$$$
$$$m = 4$$$	$$$0$$$	$$$0$$$	$$$12$$$	$$$12$$$
$$$m = 5$$$	$$$0$$$	$$$4$$$	$$$16$$$	$$$4$$$

The answer for $$$(m, k) = (5, 1)$$$ is $$$4$$$. In fact, there are $$$4$$$ permutations of length $$$4$$$ such that $$$a_i + a_{i+1} \geq 5$$$ exactly $$$1$$$ time: $$$[2, 1, 3, 4]$$$, $$$[3, 1, 2, 4]$$$, $$$[4, 2, 1, 3]$$$, $$$[4, 3, 1, 2]$$$.

In the third test case, the answers for all $$$(m, k)$$$ are shown in the following table:

	$$$k = 0$$$	$$$k = 1$$$	$$$k = 2$$$	$$$k = 3$$$	$$$k = 4$$$	$$$k = 5$$$	$$$k = 6$$$	$$$k = 7$$$
$$$m = 3$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$40320$$$
$$$m = 4$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$10080$$$	$$$30240$$$
$$$m = 5$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$1440$$$	$$$17280$$$	$$$21600$$$
$$$m = 6$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$480$$$	$$$8640$$$	$$$21600$$$	$$$9600$$$
$$$m = 7$$$	$$$0$$$	$$$0$$$	$$$0$$$	$$$96$$$	$$$3456$$$	$$$16416$$$	$$$16896$$$	$$$3456$$$
$$$m = 8$$$	$$$0$$$	$$$0$$$	$$$48$$$	$$$2160$$$	$$$12960$$$	$$$18240$$$	$$$6480$$$	$$$432$$$
$$$m = 9$$$	$$$0$$$	$$$16$$$	$$$1152$$$	$$$9648$$$	$$$18688$$$	$$$9648$$$	$$$1152$$$	$$$16$$$

Short Permutation Problem

Examples