




def     __magic_name__       (				A	:      int     ):






			'''simple docstring'''







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

			a					=    [0] * (upper_limit + 1)

			# Base case: C(0) = C(1) = 1
			a					=    1
			if upper_limit > 0:
						a					=    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(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:
						__lowerCAmelCase	:       Any     =	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 math

import flax.linen as nn
import jax.numpy as jnp




def     __magic_name__       (				A	:      jnp.ndarray,						A	:      int,						A	:      float = 1,						A	:      float = 1,						A	:      float = 1.0E4,						A	:      bool = False,						A	:      float = 1.0,						):






			'''simple docstring'''







			assert timesteps.ndim == 1, "Timesteps should be a 1d-array"
			assert embedding_dim % 2 == 0, F"""Embedding dimension {embedding_dim} should be even"""
			a					=    float(embedding_dim // 2     )
			a					=    math.log(max_timescale / min_timescale     ) / (num_timescales - freq_shift)
			a					=    min_timescale * jnp.exp(jnp.arange(A,						dtype=jnp.floataa     ) * -log_timescale_increment     )
			a					=    jnp.expand_dims(A,						1     ) * jnp.expand_dims(A,						0     )

			# scale embeddings
			a					=    scale * emb

			if flip_sin_to_cos:
						a					=    jnp.concatenate([jnp.cos(A     ), jnp.sin(A     )],						axis=1     )
			else:
						a					=    jnp.concatenate([jnp.sin(A     ), jnp.cos(A     )],						axis=1     )
			a					=    jnp.reshape(A,						[jnp.shape(A     )[0], embedding_dim]     )
			return signal




class 					snake_case__							(nn.Module     ):
				"""simple docstring"""



				SCREAMING_SNAKE_CASE_ :							int											=      32
				SCREAMING_SNAKE_CASE_ :							jnp.dtype											=      jnp.floataa





				@nn.compact
				def __call__(   self       :  Tuple      ,   __lowerCamelCase       :  Optional[Any]							)      ->  List[Any]:
							a					=    nn.Dense(self.time_embed_dim      ,   dtype=self.dtype      ,   name="linear_1"							)(__lowerCamelCase							)
							a					=    nn.silu(__lowerCamelCase							)
							a					=    nn.Dense(self.time_embed_dim      ,   dtype=self.dtype      ,   name="linear_2"							)(__lowerCamelCase							)
							return temb




class 					snake_case__							(nn.Module     ):
				"""simple docstring"""



				SCREAMING_SNAKE_CASE_ :							int											=      32
				SCREAMING_SNAKE_CASE_ :							bool											=      False
				SCREAMING_SNAKE_CASE_ :							float											=      1





				@nn.compact
				def __call__(   self       :  Tuple      ,   __lowerCamelCase       :  int							)      ->  Union[str, Any]:
							return get_sinusoidal_embeddings(
							    __lowerCamelCase      ,   embedding_dim=self.dim      ,   flip_sin_to_cos=self.flip_sin_to_cos      ,   freq_shift=self.freq_shift							)






