Pinely Round 3 (Div. 1 + Div. 2) |
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Finished |

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combinatorics

dp

fft

math

*1900

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I. Short Permutation Problem

time limit per test

7 secondsmemory limit per test

1024 megabytesinput

standard inputoutput

standard outputYou 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 $$$n$$$, $$$x$$$ ($$$2 \leq n \leq 4000$$$, $$$1 \leq x < 1\,000\,000\,007$$$).

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 $$$1\,000\,000\,007$$$.

Examples

Input

3 2

Output

77824

Input

4 1000000000

Output

30984329

Input

8 327869541

Output

85039220

Input

4000 1149333

Output

584870166

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:

$$$k = 0$$$ | $$$k = 1$$$ | $$$k = 2$$$ | $$$k = 3$$$ | $$$k = 4$$$ | $$$k = 5$$$ | $$$k = 6$$$ | $$$k = 7$$$ | |

$$$m = 3$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ | $$$40320$$$ |

$$$m = 4$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ | $$$10080$$$ | $$$30240$$$ |

$$$m = 5$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ | $$$1440$$$ | $$$17280$$$ | $$$21600$$$ |

$$$m = 6$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ | $$$480$$$ | $$$8640$$$ | $$$21600$$$ | $$$9600$$$ |

$$$m = 7$$$ | $$$0$$$ | $$$0$$$ | $$$0$$$ | $$$96$$$ | $$$3456$$$ | $$$16416$$$ | $$$16896$$$ | $$$3456$$$ |

$$$m = 8$$$ | $$$0$$$ | $$$0$$$ | $$$48$$$ | $$$2160$$$ | $$$12960$$$ | $$$18240$$$ | $$$6480$$$ | $$$432$$$ |

$$$m = 9$$$ | $$$0$$$ | $$$16$$$ | $$$1152$$$ | $$$9648$$$ | $$$18688$$$ | $$$9648$$$ | $$$1152$$$ | $$$16$$$ |

Codeforces (c) Copyright 2010-2024 Mike Mirzayanov

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