Problem p03365 - Generation 3

Orig Description

Score : 800 points
Problem StatementThere are N squares lining up in a row, numbered 1 through N from left to right.
Initially, all squares are white.
We also have N-1 painting machines, numbered 1 through N-1.
When operated, Machine i paints Square i and i+1 black.
Snuke will operate these machines one by one.
The order in which he operates them is represented by a permutation of (1, 2, ..., N-1), P, which means that the i-th operated machine is Machine P_i.
Here, the score of a permutation P is defined as the number of machines that are operated before all the squares are painted black for the first time, when the machines are operated in the order specified by P.
Snuke has not decided what permutation P to use, but he is interested in the scores of possible permutations.
Find the sum of the scores over all possible permutations for him.
Since this can be extremely large, compute the sum modulo 10^9+7.
Constraints
2 \leq N \leq 10^6
InputInput is given from Standard Input in the following format:
N
OutputPrint the sum of the scores over all possible permutations, modulo 10^9+7.
Sample Input 14
Sample Output 116
There are six possible permutations as P.
Among them, P = (1, 3, 2) and P = (3, 1, 2) have a score of 2, and the others have a score of 3.
Thus, the answer is 2 \times 2 + 3 \times 4 = 16.
Sample Input 22
Sample Output 21
There is only one possible permutation: P = (1), which has a score of 1.
Sample Input 35
Sample Output 384
Sample Input 4100000
Sample Output 4341429644

Extracted Specification

An integer \(N\) (2 ≤ \(N\) ≤ 10^6), representing some quantity or size.

### Example Input:

```
4
```

### Function Signature:
Write a function f(N) that takes in the input.
```python
def f(N: int):
    '''
    N: an integer
    '''
```

Test Case Generator

Sure, I can help you generate the test cases for the given problem. The test case generator will generate random value for \( N \) within the given constraints and will compute the output based on the problem statement. Here is the test case generator for the given problem.

```python
import random

def tcgen():
    # Generate random N within the given constraints
    N = random.randint(2, 10**6)
    
    return N

def main():
    # Generate and print a test case
    N = tcgen()
    print(N)

if __name__ == "__main__":
    main()
```

This script generates a random integer \( N \) within the range \([2, 10^6]\) and prints it, which represents the input for the problem. Note that because the problem involves calculating the sum of scores over all possible permutations, actually generating some output based on it would require a more detailed implementation and understanding of the algorithm for calculation.

For the provided competitive programming problem, computing the expected output involves deeper combinatorial logic and handling large numbers efficiently, often beyond the scope of simple random test generation. However, for competitive programming, typically generating the input as shown above is sufficient, while the solution implementation handles the output generation.

Extract Arguments

def extract_arguments(fh):
    N = int(fh.readline().strip())
    return N