Sum of Products of Subarrays

Write a Python function `sum_of_subarray_products` that computes the sum of the products of all possible subarrays of a given list of integers.

A subarray is a contiguous portion of the array. For example, for the array `[1, 2, 3]`, the subarrays are `[1]`, `[2]`, `[3]`, `[1, 2]`, `[2, 3]`, and `[1, 2, 3]`.

#### Example Usage:

```python [main.nopy]
print(sum_of_subarray_products([1, 2, 3]))  # Output: 20
print(sum_of_subarray_products([1, 2]))     # Output: 5
print(sum_of_subarray_products([1, 2, 3, 4])) # Output: 84
```

#### Constraints:

- The input list will contain at least one integer.
- The integers in the list will be non-negative.

Implement the function efficiently.
def sum_of_subarray_products(arr):
    """
    Calculate the sum of the products of all possible subarrays of the given list.

    Args:
        arr (list): A list of integers.

    Returns:
        int: The sum of the products of all possible subarrays.
    """
    # Placeholder for the solution
    pass
Advanced math
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