



def     SCREAMING_SNAKE_CASE(      __UpperCamelCase )	->						"list[int]":
				if upper_limit < 0:
								raise ValueError("Limit for the Catalan sequence must be ≥ 0" )

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

				# Base case: C(0) = C(1) = 1
				a__	:							Union[str, Any]      						= 1
				if upper_limit > 0:
								a__	:							Tuple      						= 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(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()







import qiskit






def     SCREAMING_SNAKE_CASE(      __UpperCamelCase = 2 )	->						qiskit.result.counts.Counts:
				a__	:							Any      						= qubits

				# Using Aer's simulator
				a__	:							Dict      						= qiskit.Aer.get_backend("aer_simulator" )

				# Creating a Quantum Circuit acting on the q register
				a__	:							Dict      						= qiskit.QuantumCircuit(__UpperCamelCase	,				__UpperCamelCase )

				# Adding a H gate on qubit 0 (now q0 in superposition)
				circuit.h(0 )

				for i in range(1	,				__UpperCamelCase ):
								# Adding CX (CNOT) gate
								circuit.cx(i - 1	,				__UpperCamelCase )

				# Mapping the quantum measurement to the classical bits
				circuit.measure(list(range(__UpperCamelCase ) )	,				list(range(__UpperCamelCase ) ) )

				# Now measuring any one qubit would affect other qubits to collapse
				# their super position and have same state as the measured one.

				# Executing the circuit on the simulator
				a__	:							Any      						= qiskit.execute(__UpperCamelCase	,				__UpperCamelCase	,				shots=10_00 )

				return job.result().get_counts(__UpperCamelCase )


if __name__ == "__main__":
						print(F'Total count for various states are: {quantum_entanglement(3)}')


