





def       SCREAMING_SNAKE_CASE__  (       _lowercase     : Optional[Any]			)   ->							"list[int]":




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

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

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







import operator as op


lowerCamelCase		       =      """scaler.pt"""
lowerCamelCase		       =      """pytorch_model"""
lowerCamelCase		       =      """random_states"""
lowerCamelCase		       =      """optimizer"""
lowerCamelCase		       =      """scheduler"""
lowerCamelCase		       =      """pytorch_model.bin"""
lowerCamelCase		       =      """pytorch_model.bin.index.json"""
lowerCamelCase		       =      """model.safetensors"""
lowerCamelCase		       =      """model.safetensors.index.json"""
lowerCamelCase		       =      """1.10.2"""
lowerCamelCase		       =      """py38"""
lowerCamelCase		       =      """4.17.0"""
lowerCamelCase		       =      ["""ml.p3.16xlarge""", """ml.p3dn.24xlarge""", """ml.p4dn.24xlarge"""]
lowerCamelCase		       =      ["""FULL_SHARD""", """SHARD_GRAD_OP""", """NO_SHARD""", """HYBRID_SHARD""", """HYBRID_SHARD_ZERO2"""]
lowerCamelCase		       =      ["""TRANSFORMER_BASED_WRAP""", """SIZE_BASED_WRAP""", """NO_WRAP"""]
lowerCamelCase		       =      ["""BACKWARD_PRE""", """BACKWARD_POST""", """NO_PREFETCH"""]
lowerCamelCase		       =      ["""FULL_STATE_DICT""", """LOCAL_STATE_DICT""", """SHARDED_STATE_DICT"""]
lowerCamelCase		       =      """2.0.1"""
lowerCamelCase		       =      ["""pdsh""", """standard""", """openmpi""", """mvapich"""]
lowerCamelCase		       =      ["""default""", """reduce-overhead""", """max-autotune"""]

lowerCamelCase		       =      {""">""": op.gt, """>=""": op.ge, """==""": op.eq, """!=""": op.ne, """<=""": op.le, """<""": op.lt}

# These are the args for `torch.distributed.launch` for pytorch < 1.9
lowerCamelCase		       =      [
    """nnodes""",
    """nproc_per_node""",
    """rdzv_backend""",
    """rdzv_endpoint""",
    """rdzv_id""",
    """rdzv_conf""",
    """standalone""",
    """max_restarts""",
    """monitor_interval""",
    """start_method""",
    """role""",
    """module""",
    """m""",
    """no_python""",
    """run_path""",
    """log_dir""",
    """r""",
    """redirects""",
    """t""",
    """tee""",
    """node_rank""",
    """master_addr""",
    """master_port""",
]

lowerCamelCase		       =      ["""DEEPSPEED""", """MULTI_GPU""", """FSDP""", """MEGATRON_LM"""]
lowerCamelCase		       =      ["""DEEPSPEED""", """MULTI_XPU""", """FSDP"""]


