






def 							lowerCamelCase__ (       __lowerCamelCase     :	int						):







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

  __UpperCAmelCase				:      Optional[Any]	 =    [0] * (upper_limit + 1)

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






_lowercase							:   Dict						=						"0.21.0"

from .accelerator import Accelerator
from .big_modeling import (
    cpu_offload,
    cpu_offload_with_hook,
    disk_offload,
    dispatch_model,
    init_empty_weights,
    init_on_device,
    load_checkpoint_and_dispatch,
)
from .data_loader import skip_first_batches
from .launchers import debug_launcher, notebook_launcher
from .state import PartialState
from .utils import (
    DeepSpeedPlugin,
    DistributedDataParallelKwargs,
    DistributedType,
    FullyShardedDataParallelPlugin,
    GradScalerKwargs,
    InitProcessGroupKwargs,
    find_executable_batch_size,
    infer_auto_device_map,
    is_rich_available,
    load_checkpoint_in_model,
    synchronize_rng_states,
)


if is_rich_available():
				from .utils import rich



