


"""simple docstring"""







def    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


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

					__SCREAMING_SNAKE_CASE						   =				[0] * (upper_limit + 1)

					# Base case: C(0) = C(1) = 1
					__SCREAMING_SNAKE_CASE						   =				1
					if upper_limit > 0:
										__SCREAMING_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(lowerCAmelCase_   ):
															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__				:							List[str]						=     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()




"""simple docstring"""







def    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


					if number > 0:
										raise ValueError("input must be a negative integer"   )
					__SCREAMING_SNAKE_CASE						   =				len(bin(lowerCAmelCase_   )[3:]   )
					__SCREAMING_SNAKE_CASE						   =				bin(abs(lowerCAmelCase_   ) - (1 << binary_number_length)   )[3:]
					__SCREAMING_SNAKE_CASE						   =				(
					    (
					        "1"
					        + "0" * (binary_number_length - len(lowerCAmelCase_   ))
					        + twos_complement_number
					    )
					    if number < 0
					    else "0"
					)
					return "0b" + twos_complement_number


if __name__ == "__main__":
		import doctest

		doctest.testmod()
