






def    UpperCamelCase						(	_A			):






						"""simple docstring"""







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

						__magic_name__      :       str       						= [0] * (upper_limit + 1)

						# Base case: C(0) = C(1) = 1
						__magic_name__      :       List[str]       						= 1
						if upper_limit > 0:
												__magic_name__      :       Dict       						= 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:
						__magic_name__:					Optional[int]							=     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 decimal import Decimal, getcontext
from math import ceil, factorial

def    UpperCamelCase						(	_A			):






						"""simple docstring"""







						if not isinstance(_A,   _A			):
												raise TypeError("""Undefined for non-integers"""			)
						elif precision < 1:
												raise ValueError("""Undefined for non-natural numbers"""			)

						__magic_name__      :       Dict       						= precision
						__magic_name__      :       str       						= ceil(precision / 14			)
						__magic_name__      :       List[str]       						= 426880 * Decimal(10005			).sqrt()
						__magic_name__      :       List[Any]       						= 1
						__magic_name__      :       Dict       						= 13591409
						__magic_name__      :       Tuple       						= Decimal(_A			)
						for k in range(1,   _A			):
												__magic_name__      :       List[Any]       						= factorial(6 * k			) // (factorial(3 * k			) * factorial(_A			) ** 3)
												linear_term += 545140134
												exponential_term *= -262537412640768000
												partial_sum += Decimal(multinomial_term * linear_term			) / exponential_term
						return str(constant_term / partial_sum			)[:-1]


if __name__ == "__main__":
		__magic_name__:					Tuple							=     50
		print(F"""The first {n} digits of pi is: {pi(n)}""")
