


def 					lowerCamelCase__						(   a						: int					)     ->						"list[int]":





			"""simple docstring"""

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

			a__			:Optional[int]    	=     [0] * (upper_limit + 1)

			# Base case: C(0) = C(1) = 1
			a__			:Union[str, Any]    	=     1
			if upper_limit > 0:
						a__			:Union[str, Any]    	=     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(__lowercase					):
									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:
						snake_case__      			=    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 __future__ import annotations



def       _SCREAMING_SNAKE_CASE    (			__lowercase :      list[int]      ,					__lowercase :      int     )    ->   bool:





    """simple docstring"""

    if len(__lowercase     ) == 0:
        return False
    __A       =						len(__lowercase     ) // 2
    if a_list[midpoint] == item:
        return True
    if item < a_list[midpoint]:
        return binary_search(a_list[:midpoint]      ,					__lowercase     )
    else:
        return binary_search(a_list[midpoint + 1 :]      ,					__lowercase     )


if __name__ == "__main__":
  __a   :							Tuple						= input("Enter numbers separated by comma:\n").strip()
  __a   :							Any						= [int(item.strip()) for item in user_input.split(",")]
  __a   :							List[Any]						= int(input("Enter the number to be found in the list:\n").strip())
  __a   :							Optional[int]						= "" if binary_search(sequence, target) else "not "
  print(f"""{target} was {not_str}found in {sequence}""")




