#### Bell Numbers

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```"""
Bell numbers represent the number of ways to partition a set into non-empty
subsets. This module provides functions to calculate Bell numbers for sets of
integers. In other words, the first (n + 1) Bell numbers.

https://en.wikipedia.org/wiki/Bell_number
"""

def bell_numbers(max_set_length: int) -> list[int]:
"""
Calculate Bell numbers for the sets of lengths from 0 to max_set_length.
In other words, calculate first (max_set_length + 1) Bell numbers.

Args:
max_set_length (int): The maximum length of the sets for which
Bell numbers are calculated.

Returns:
list: A list of Bell numbers for sets of lengths from 0 to max_set_length.

Examples:
>>> bell_numbers(0)

>>> bell_numbers(1)
[1, 1]
>>> bell_numbers(5)
[1, 1, 2, 5, 15, 52]
"""
if max_set_length < 0:
raise ValueError("max_set_length must be non-negative")

bell =  * (max_set_length + 1)
bell = 1

for i in range(1, max_set_length + 1):
for j in range(i):
bell[i] += _binomial_coefficient(i - 1, j) * bell[j]

return bell

def _binomial_coefficient(total_elements: int, elements_to_choose: int) -> int:
"""
Calculate the binomial coefficient C(total_elements, elements_to_choose)

Args:
total_elements (int): The total number of elements.
elements_to_choose (int): The number of elements to choose.

Returns:
int: The binomial coefficient C(total_elements, elements_to_choose).

Examples:
>>> _binomial_coefficient(5, 2)
10
>>> _binomial_coefficient(6, 3)
20
"""
if elements_to_choose in {0, total_elements}:
return 1

if elements_to_choose > total_elements - elements_to_choose:
elements_to_choose = total_elements - elements_to_choose

coefficient = 1
for i in range(elements_to_choose):
coefficient *= total_elements - i
coefficient //= i + 1

return coefficient

if __name__ == "__main__":
import doctest

doctest.testmod()
```  