
def 						lowercase_      (       A__		)							->							"list[int]":
    """simple docstring"""

    if upper_limit < 0:
        raise ValueError("Limit for the Catalan sequence must be ≥ 0"		)

    snake_case								= [0] * (upper_limit + 1)

    # Base case: C(0) = C(1) = 1
    snake_case								= 1
    if upper_limit > 0:
        snake_case								= 1

    # Recurrence relation: C(i) = sum(C(j).C(i-j-1)), from j = 0 to i
    for i in range(2			,		upper_limit + 1		):
        for j in range(A__		):
            catalan_list[i] += catalan_list[j] * catalan_list[i - j - 1]

    return catalan_list


if __name__ == "__main__":
   print("\n********* Catalan Numbers Using Dynamic Programming ************\n")
   print("\n*** Enter -1 at any time to quit ***")
   print("\nEnter the upper limit (≥ 0) for the Catalan number sequence: ", end="")
   try:
      while True:
         _A   =			int(input().strip())
         if N < 0:
            print("\n********* Goodbye!! ************")
            break
         else:
            print(f"The Catalan numbers from 0 through {N} are:")
            print(catalan_numbers(N))
            print("Try another upper limit for the sequence: ", end="")
   except (NameError, ValueError):
      print("\n********* Invalid input, goodbye! ************\n")

   import doctest

   doctest.testmod()



from operator import delitem, getitem, setitem

import pytest

from data_structures.hashing.hash_map import HashMap



def 						lowercase_      (       A__		)							->							str:
    """simple docstring"""

    return getitem, k



def 						lowercase_      (       A__			,		A__		)							->							str:
    """simple docstring"""

    return setitem, k, v



def 						lowercase_      (       A__		)							->							List[Any]:
    """simple docstring"""

    return delitem, k



def 						lowercase_      (       A__			,		A__			,		*A__		)							->							str:
    """simple docstring"""

    try:
        return fun(A__			,		*A__		), None
    except Exception as e:
        return None, e


_A   =			(
    _set("key_a", "val_a"),
    _set("key_b", "val_b"),
)

_A   =			[
    _set("key_a", "val_a"),
    _set("key_a", "val_b"),
]

_A   =			[
    _set("key_a", "val_a"),
    _set("key_b", "val_b"),
    _del("key_a"),
    _del("key_b"),
    _set("key_a", "val_a"),
    _del("key_a"),
]

_A   =			[
    _get("key_a"),
    _del("key_a"),
    _set("key_a", "val_a"),
    _del("key_a"),
    _del("key_a"),
    _get("key_a"),
]

_A   =			[
    *[_set(x, x) for x in range(5)],  # guaranteed upsize
]

_A   =			[
    *[_set(x, x) for x in range(5)],  # guaranteed upsize
    *[_del(x) for x in range(5)],
    _set("key_a", "val_b"),
]



@pytest.mark.parametrize(
    "operations"			,		(
        pytest.param(_add_items			,		id="add items"		),
        pytest.param(_overwrite_items			,		id="overwrite items"		),
        pytest.param(_delete_items			,		id="delete items"		),
        pytest.param(_access_absent_items			,		id="access absent items"		),
        pytest.param(_add_with_resize_up			,		id="add with resize up"		),
        pytest.param(_add_with_resize_down			,		id="add with resize down"		),
    )			,		)
def 						lowercase_      (       A__		)							->							List[Any]:
    """simple docstring"""

    snake_case								= HashMap(initial_block_size=4		)
    snake_case								= {}
    for _, (fun, *args) in enumerate(A__		):
        snake_case					,							snake_case								= _run_operation(A__			,		A__			,		*A__		)
        snake_case					,							snake_case								= _run_operation(A__			,		A__			,		*A__		)
        assert my_res == py_res
        assert str(A__		) == str(A__		)
        assert set(A__		) == set(A__		)
        assert len(A__		) == len(A__		)
        assert set(my.items()		) == set(py.items()		)





def 						lowercase_      (       )							->							Optional[int]:
    """simple docstring"""

    def is_public(A__		) -> bool:
        return not name.startswith("_"		)

    snake_case								= {name for name in dir({}		) if is_public(A__		)}
    snake_case								= {name for name in dir(HashMap()		) if is_public(A__		)}

    assert dict_public_names > hash_public_names

