#### Product Sum

R
```"""
Calculate the Product Sum from a Special Array.
reference: https://dev.to/sfrasica/algorithms-product-sum-from-an-array-dc6

Python doctests can be run with the following command:
python -m doctest -v product_sum.py

Calculate the product sum of a "special" array which can contain integers or nested
arrays. The product sum is obtained by adding all elements and multiplying by their
respective depths.

For example, in the array [x, y], the product sum is (x + y). In the array [x, [y, z]],
the product sum is x + 2 * (y + z). In the array [x, [y, [z]]],
the product sum is x + 2 * (y + 3z).

Example Input:
[5, 2, [-7, 1], 3, [6, [-13, 8], 4]]
Output: 12

"""

def product_sum(arr: list[int | list], depth: int) -> int:
"""
Recursively calculates the product sum of an array.

The product sum of an array is defined as the sum of its elements multiplied by
their respective depths.  If an element is a list, its product sum is calculated
recursively by multiplying the sum of its elements with its depth plus one.

Args:
arr: The array of integers and nested lists.
depth: The current depth level.

Returns:
int: The product sum of the array.

Examples:
>>> product_sum([1, 2, 3], 1)
6
>>> product_sum([-1, 2, [-3, 4]], 2)
8
>>> product_sum([1, 2, 3], -1)
-6
>>> product_sum([1, 2, 3], 0)
0
>>> product_sum([1, 2, 3], 7)
42
>>> product_sum((1, 2, 3), 7)
42
>>> product_sum({1, 2, 3}, 7)
42
>>> product_sum([1, -1], 1)
0
>>> product_sum([1, -2], 1)
-1
>>> product_sum([-3.5, [1, [0.5]]], 1)
1.5

"""
total_sum = 0
for ele in arr:
total_sum += product_sum(ele, depth + 1) if isinstance(ele, list) else ele

def product_sum_array(array: list[int | list]) -> int:
"""
Calculates the product sum of an array.

Args:
array (List[Union[int, List]]): The array of integers and nested lists.

Returns:
int: The product sum of the array.

Examples:
>>> product_sum_array([1, 2, 3])
6
>>> product_sum_array([1, [2, 3]])
11
>>> product_sum_array([1, [2, [3, 4]]])
47
>>> product_sum_array([0])
0
>>> product_sum_array([-3.5, [1, [0.5]]])
1.5
>>> product_sum_array([1, -2])
-1

"""
return product_sum(array, 1)

if __name__ == "__main__":
import doctest

doctest.testmod()
```