


import math
from typing import List, Optional, Tuple, Union

import numpy as np
import torch

from ..configuration_utils import ConfigMixin, register_to_config
from .scheduling_utils import SchedulerMixin, SchedulerOutput







class        snake_case_   (snake_case__	,						snake_case__      ):
					UpperCAmelCase__					:    Optional[Any]          =			1
					@register_to_config
					def __init__(      self   :Optional[int]			,__snake_case   :int = 10_00			,__snake_case   :Optional[Union[np.ndarray, List[float]]] = None      )					->    int:
												# set `betas`, `alphas`, `timesteps`
												self.set_timesteps(__snake_case      )

												# standard deviation of the initial noise distribution
												a__             =     1.0

												# For now we only support F-PNDM, i.e. the runge-kutta method
												# For more information on the algorithm please take a look at the paper: https://arxiv.org/pdf/2202.09778.pdf
												# mainly at formula (9), (12), (13) and the Algorithm 2.
												a__             =     4

												# running values
												a__             =     []
					def  lowerCamelCase__(      self   :Optional[int]			,__snake_case   :int			,__snake_case   :Union[str, torch.device] = None      )					->    Optional[Any]:
												a__             =     num_inference_steps
												a__             =     torch.linspace(1			,0			,num_inference_steps + 1      )[:-1]
												a__             =     torch.cat([steps, torch.tensor([0.0]      )]      )

												if self.config.trained_betas is not None:
																			a__             =     torch.tensor(self.config.trained_betas			,dtype=torch.floataa      )
												else:
																			a__             =     torch.sin(steps * math.pi / 2      ) ** 2

												a__             =     (1.0 - self.betas**2) ** 0.5

												a__             =     (torch.atana(self.betas			,self.alphas      ) / math.pi * 2)[:-1]
												a__             =     timesteps.to(__snake_case      )

												a__             =     []
					def  lowerCamelCase__(      self   :Optional[int]			,__snake_case   :torch.FloatTensor			,__snake_case   :int			,__snake_case   :torch.FloatTensor			,__snake_case   :bool = True			,)					->    List[Any]:
												if self.num_inference_steps is None:
																			raise ValueError(
																			    'Number of inference steps is \'None\', you need to run \'set_timesteps\' after creating the scheduler'      )

												a__             =     (self.timesteps == timestep).nonzero().item()
												a__             =     timestep_index + 1

												a__             =     sample * self.betas[timestep_index] + model_output * self.alphas[timestep_index]
												self.ets.append(__snake_case      )

												if len(self.ets      ) == 1:
																			a__             =     self.ets[-1]
												elif len(self.ets      ) == 2:
																			a__             =     (3 * self.ets[-1] - self.ets[-2]) / 2
												elif len(self.ets      ) == 3:
																			a__             =     (23 * self.ets[-1] - 16 * self.ets[-2] + 5 * self.ets[-3]) / 12
												else:
																			a__             =     (1 / 24) * (55 * self.ets[-1] - 59 * self.ets[-2] + 37 * self.ets[-3] - 9 * self.ets[-4])

												a__             =     self._get_prev_sample(__snake_case			,__snake_case			,__snake_case			,__snake_case      )

												if not return_dict:
																			return (prev_sample,)

												return SchedulerOutput(prev_sample=__snake_case      )
					def  lowerCamelCase__(      self   :Union[str, Any]			,__snake_case   :torch.FloatTensor			,*__snake_case   :Union[str, Any]			,**__snake_case   :Dict      )					->    Any:
												return sample
					def  lowerCamelCase__(      self   :Optional[Any]			,__snake_case   :Optional[int]			,__snake_case   :int			,__snake_case   :Optional[Any]			,__snake_case   :List[str]      )					->    Any:
												a__             =     self.alphas[timestep_index]
												a__             =     self.betas[timestep_index]

												a__             =     self.alphas[prev_timestep_index]
												a__             =     self.betas[prev_timestep_index]

												a__             =     (sample - sigma * ets) / max(__snake_case			,1E-8      )
												a__             =     next_alpha * pred + ets * next_sigma

												return prev_sample



					def __len__(      self   :Union[str, Any]      )					->    int:
												return self.config.num_train_timesteps





'''simple docstring'''




def 	UpperCamelCase_							(					_UpperCAmelCase  :      int			,	_UpperCAmelCase  :      int    )    ->    int:


       """simple docstring"""







       _UpperCAmelCase     :  List[str]							      =    1  # To kept the Calculated Value
       # Since C(n, k) = C(n, n-k)
       if k > (n - k):
              _UpperCAmelCase     :  Any							      =    n - k
       # Calculate C(n,k)
       for i in range(_UpperCAmelCase    ):
              result *= n - i
              result //= i + 1
       return result

def 	UpperCamelCase_							(					_UpperCAmelCase  :      int    )    ->    int:


       """simple docstring"""







       return binomial_coefficient(2 * node_count			,	_UpperCAmelCase    ) // (node_count + 1)

def 	UpperCamelCase_							(					_UpperCAmelCase  :      int    )    ->    int:


       """simple docstring"""







       if n < 0:
              raise ValueError("factorial() not defined for negative values"    )
       _UpperCAmelCase     :  List[str]							      =    1
       for i in range(1			,	n + 1    ):
              result *= i
       return result

def 	UpperCamelCase_							(					_UpperCAmelCase  :      int    )    ->    int:


       """simple docstring"""







       return catalan_number(_UpperCAmelCase    ) * factorial(_UpperCAmelCase    )


if __name__ == "__main__":
  __SCREAMING_SNAKE_CASE       :     Dict     =  int(input("""Enter the number of nodes: """).strip() or 0)
  if node_count <= 0:
    raise ValueError("""We need some nodes to work with.""")
  print(
      F'Given {node_count} nodes, there are {binary_tree_count(node_count)} '
      F'binary trees and {catalan_number(node_count)} binary search trees.'
  )

