'''simple docstring'''


def       lowerCAmelCase					(		UpperCamelCase__				:   int       ):


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

	__UpperCAmelCase							=			[0] * (upper_limit + 1)

	# Base case: C(0) = C(1) = 1
	__UpperCAmelCase							=			1
	if upper_limit > 0:
		__UpperCAmelCase							=			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(UpperCamelCase__       ):
			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:
															__lowerCAmelCase	:       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()

'''simple docstring'''


from __future__ import annotations

import math





def       lowerCAmelCase					(		UpperCamelCase__				:   float    ,					UpperCamelCase__				:   int       ):


	"""simple docstring"""
	__UpperCAmelCase							=			u
	for i in range(1    ,					UpperCamelCase__       ):
		__UpperCAmelCase							=			temp * (u - i)
	return temp





def       lowerCAmelCase					(		):


	"""simple docstring"""
	__UpperCAmelCase							=			int(input('''enter the numbers of values: '''       )       )
	__UpperCAmelCase							=			[]
	for _ in range(UpperCamelCase__       ):
		y.append([]       )
	for i in range(UpperCamelCase__       ):
		for j in range(UpperCamelCase__       ):
			y[i].append(UpperCamelCase__       )
			__UpperCAmelCase							=			0

	print('''enter the values of parameters in a list: '''       )
	__UpperCAmelCase							=			list(map(UpperCamelCase__    ,					input().split()       )       )

	print('''enter the values of corresponding parameters: '''       )
	for i in range(UpperCamelCase__       ):
		__UpperCAmelCase							=			float(input()       )

	__UpperCAmelCase							=			int(input('''enter the value to interpolate: '''       )       )
	__UpperCAmelCase							=			(value - x[0]) / (x[1] - x[0])

	# for calculating forward difference table

	for i in range(1    ,					UpperCamelCase__       ):
		for j in range(n - i       ):
			__UpperCAmelCase							=			y[j + 1][i - 1] - y[j][i - 1]

	__UpperCAmelCase							=			y[0][0]
	for i in range(1    ,					UpperCamelCase__       ):
		summ += (ucal(UpperCamelCase__    ,					UpperCamelCase__       ) * y[0][i]) / math.factorial(UpperCamelCase__       )

	print(f"""the value at {value} is {summ}"""       )


if __name__ == "__main__":
					main()
