Certainly! Here is a Python function to generate test cases for the given problem statement:

```python
import random

def tcgen():
    # Generate random counts for each type of tetromino
    a_I = random.randint(0, 10**9)
    a_O = random.randint(0, 10**9)
    a_T = random.randint(0, 10**9)
    a_J = random.randint(0, 10**9)
    a_L = random.randint(0, 10**9)
    a_S = random.randint(0, 10**9)
    a_Z = random.randint(0, 10**9)
    
    # Ensure that there is at least one tetromino
    if a_I + a_O + a_T + a_J + a_L + a_S + a_Z == 0:
        tetromino_type = random.choice(['a_I', 'a_O', 'a_T', 'a_J', 'a_L', 'a_S', 'a_Z'])
        if tetromino_type == 'a_I':
            a_I = 1
        elif tetromino_type == 'a_O':
            a_O = 1
        elif tetromino_type == 'a_T':
            a_T = 1
        elif tetromino_type == 'a_J':
            a_J = 1
        elif tetromino_type == 'a_L':
            a_L = 1
        elif tetromino_type == 'a_S':
            a_S = 1
        elif tetromino_type == 'a_Z':
            a_Z = 1

    # Return the generated test case
    return a_I, a_O, a_T, a_J, a_L, a_S, a_Z

# Example usage:
for _ in range(5):
    print(tcgen())
```

This function generates random values for each of the tetromino counts within the given constraints. It ensures that the sum of all the tetromino counts is at least one by setting at least one of the tetromino counts to 1 if they all sum to zero. The generated test cases can be used to test the solution to the problem statement.