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! Below is a test case generator for the given problem statement. This generator will produce random inputs according to the constraints provided:
```python
import random
def generate_test_case():
# Define constraints
MIN_N = 2
MAX_N = 10**6
MOD = 10**9 + 7
# Randomly determine the size of N
N = random.randint(MIN_N, MAX_N)
return N
# Example usage of the test case generator
if __name__ == "__main__":
print(generate_test_case())
```
This generator will create a random test case for the problem where \(N\) is a random integer between 2 and \(10^6\). The generated test case can then be used to test implementations of the problem.
For more complex testing, especially to test edge cases, you might want to include specific values for \(N\) such as the minimum (2) and maximum (\(10^6\)) values:
```python
def generate_edge_cases():
edge_cases = [2, 10**6]
return edge_cases
# Example usage of the edge case generator
if __name__ == "__main__":
for case in generate_edge_cases():
print(case)
```
This will output edge cases of the problem, ensuring that your implementation can handle the smallest and largest inputs as specified by the constraints.