

def 					_UpperCAmelCase       (  _SCREAMING_SNAKE_CASE					:		int       ):





    """simple docstring"""


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

    SCREAMING_SNAKE_CASE_									=						[0] * (upper_limit + 1)

    # Base case: C(0) = C(1) = 1
    SCREAMING_SNAKE_CASE_									=						1
    if upper_limit > 0:
        SCREAMING_SNAKE_CASE_									=						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(_SCREAMING_SNAKE_CASE       ):
            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__  :      Dict			=    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 typing import List

import numpy as np


def 					_UpperCAmelCase       (  _SCREAMING_SNAKE_CASE					:		dict       ):





    """simple docstring"""


    SCREAMING_SNAKE_CASE_									=						{key: len(_SCREAMING_SNAKE_CASE       ) for key, value in gen_kwargs.items() if isinstance(_SCREAMING_SNAKE_CASE ,   _SCREAMING_SNAKE_CASE       )}
    if len(set(lists_lengths.values()       )       ) > 1:
        raise RuntimeError(
            (
                'Sharding is ambiguous for this dataset: '
                + 'we found several data sources lists of different lengths, and we don\'t know over which list we should parallelize:\n'
                + '\n'.join(f"""\t- key {key} has length {length}""" for key, length in lists_lengths.items()       )
                + '\nTo fix this, check the \'gen_kwargs\' and make sure to use lists only for data sources, '
                + 'and use tuples otherwise. In the end there should only be one single list, or several lists with the same length.'
            )       )
    SCREAMING_SNAKE_CASE_									=						max(lists_lengths.values() ,   default=0       )
    return max(1 ,   _SCREAMING_SNAKE_CASE       )


def 					_UpperCAmelCase       (  _SCREAMING_SNAKE_CASE					:		int ,   _SCREAMING_SNAKE_CASE					:		int       ):





    """simple docstring"""


    SCREAMING_SNAKE_CASE_									=						[]
    for group_idx in range(_SCREAMING_SNAKE_CASE       ):
        SCREAMING_SNAKE_CASE_									=						num_shards // max_num_jobs + (group_idx < (num_shards % max_num_jobs))
        if num_shards_to_add == 0:
            break
        SCREAMING_SNAKE_CASE_									=						shards_indices_per_group[-1].stop if shards_indices_per_group else 0
        SCREAMING_SNAKE_CASE_									=						range(_SCREAMING_SNAKE_CASE ,   start + num_shards_to_add       )
        shards_indices_per_group.append(_SCREAMING_SNAKE_CASE       )
    return shards_indices_per_group


def 					_UpperCAmelCase       (  _SCREAMING_SNAKE_CASE					:		dict ,   _SCREAMING_SNAKE_CASE					:		int       ):





    """simple docstring"""


    SCREAMING_SNAKE_CASE_									=						_number_of_shards_in_gen_kwargs(_SCREAMING_SNAKE_CASE       )
    if num_shards == 1:
        return [dict(_SCREAMING_SNAKE_CASE       )]
    else:
        SCREAMING_SNAKE_CASE_									=						_distribute_shards(num_shards=_SCREAMING_SNAKE_CASE ,   max_num_jobs=_SCREAMING_SNAKE_CASE       )
        return [
            {
                key: [value[shard_idx] for shard_idx in shard_indices_per_group[group_idx]]
                if isinstance(_SCREAMING_SNAKE_CASE ,   _SCREAMING_SNAKE_CASE       )
                else value
                for key, value in gen_kwargs.items()
            }
            for group_idx in range(len(_SCREAMING_SNAKE_CASE       )       )
        ]


def 					_UpperCAmelCase       (  _SCREAMING_SNAKE_CASE					:		List[dict]       ):





    """simple docstring"""


    return {
        key: [value for gen_kwargs in gen_kwargs_list for value in gen_kwargs[key]]
        if isinstance(gen_kwargs_list[0][key] ,   _SCREAMING_SNAKE_CASE       )
        else gen_kwargs_list[0][key]
        for key in gen_kwargs_list[0]
    }





def 					_UpperCAmelCase       (  _SCREAMING_SNAKE_CASE					:		np.random.Generator ,   _SCREAMING_SNAKE_CASE					:		dict       ):





    """simple docstring"""


    SCREAMING_SNAKE_CASE_									=						{len(_SCREAMING_SNAKE_CASE       ) for value in gen_kwargs.values() if isinstance(_SCREAMING_SNAKE_CASE ,   _SCREAMING_SNAKE_CASE       )}
    SCREAMING_SNAKE_CASE_									=						{}
    for size in list_sizes:
        SCREAMING_SNAKE_CASE_									=						list(range(_SCREAMING_SNAKE_CASE       )       )
        rng.shuffle(indices_per_size[size]       )
    # Now let's copy the gen_kwargs and shuffle the lists based on their sizes
    SCREAMING_SNAKE_CASE_									=						dict(_SCREAMING_SNAKE_CASE       )
    for key, value in shuffled_kwargs.items():
        if isinstance(_SCREAMING_SNAKE_CASE ,   _SCREAMING_SNAKE_CASE       ):
            SCREAMING_SNAKE_CASE_									=						[value[i] for i in indices_per_size[len(_SCREAMING_SNAKE_CASE       )]]
    return shuffled_kwargs


