



"""simple docstring"""




def   lowercase__     (			_UpperCAmelCase = 1_00				)     ->			int:






      '''simple docstring'''


      lowercase  :      List[str]					=   [[0 for _ in range(_UpperCAmelCase				)] for _ in range(m + 1				)]
      for i in range(m + 1				):
            lowercase  :      Union[str, Any]					=   1

      for n in range(m + 1				):
            for k in range(1	,      _UpperCAmelCase				):
                  memo[n][k] += memo[n][k - 1]
                  if n > k:
                        memo[n][k] += memo[n - k - 1][k]

      return memo[m][m - 1] - 1


if __name__ == "__main__":
     print(solution(int(input('Enter a number: ').strip())))





"""simple docstring"""




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

import numpy as np
import torch

from ..configuration_utils import ConfigMixin, register_to_config
from ..utils import BaseOutput, logging, randn_tensor
from .scheduling_utils import KarrasDiffusionSchedulers, SchedulerMixin


_UpperCamelCase:				str    							=  logging.get_logger(__name__)  # pylint: disable=invalid-name


@dataclass
# Copied from diffusers.schedulers.scheduling_ddpm.DDPMSchedulerOutput with DDPM->EulerAncestralDiscrete
class   a__		(						SCREAMING_SNAKE_CASE__     ):
   _lowerCamelCase												=				42
   _lowerCamelCase												=				None




def   lowercase__     (			_UpperCAmelCase	,      _UpperCAmelCase=0.9_9_9	,      _UpperCAmelCase="cosine"	,      )     ->			Optional[Any]:






      '''simple docstring'''


      if alpha_transform_type == "cosine":

            def alpha_bar_fn(_UpperCAmelCase				):
                  return math.cos((t + 0.0_0_8) / 1.0_0_8 * math.pi / 2				) ** 2

      elif alpha_transform_type == "exp":

            def alpha_bar_fn(_UpperCAmelCase				):
                  return math.exp(t * -1_2.0				)

      else:
            raise ValueError(f'''Unsupported alpha_tranform_type: {alpha_transform_type}'''				)

      lowercase  :      Any					=   []
      for i in range(_UpperCAmelCase				):
            lowercase  :      List[str]					=   i / num_diffusion_timesteps
            lowercase  :      str					=   (i + 1) / num_diffusion_timesteps
            betas.append(min(1 - alpha_bar_fn(_UpperCAmelCase				) / alpha_bar_fn(_UpperCAmelCase				)	,      _UpperCAmelCase				)				)
      return torch.tensor(_UpperCAmelCase	,      dtype=torch.floataa				)





class   a__		(						SCREAMING_SNAKE_CASE__,  SCREAMING_SNAKE_CASE__     ):
   _lowerCamelCase												=				[e.name for e in KarrasDiffusionSchedulers]
   _lowerCamelCase												=				1


   @register_to_config
   def __init__(							self      :							str,     lowerCAmelCase      :							int = 1000,     lowerCAmelCase      :							float = 0.0001,     lowerCAmelCase      :							float = 0.02,     lowerCAmelCase      :							str = "linear",     lowerCAmelCase      :							Optional[Union[np.ndarray, List[float]]] = None,     lowerCAmelCase      :							str = "epsilon",     lowerCAmelCase      :							str = "linspace",     lowerCAmelCase      :							int = 0,     )    ->       List[Any]:
         if trained_betas is not None:
               lowercase  :      Union[str, Any]					=   torch.tensor(lowerCAmelCase,     dtype=torch.floataa			)
         elif beta_schedule == "linear":
               lowercase  :      Union[str, Any]					=   torch.linspace(lowerCAmelCase,     lowerCAmelCase,     lowerCAmelCase,     dtype=torch.floataa			)
         elif beta_schedule == "scaled_linear":
               # this schedule is very specific to the latent diffusion model.
               lowercase  :      str					=   (
                   torch.linspace(beta_start**0.5,     beta_end**0.5,     lowerCAmelCase,     dtype=torch.floataa			) ** 2
               )
         elif beta_schedule == "squaredcos_cap_v2":
               # Glide cosine schedule
               lowercase  :      List[Any]					=   betas_for_alpha_bar(lowerCAmelCase			)
         else:
               raise NotImplementedError(f'''{beta_schedule} does is not implemented for {self.__class__}'''			)

         lowercase  :      Any					=   1.0 - self.betas
         lowercase  :      Optional[int]					=   torch.cumprod(self.alphas,     dim=0			)

         lowercase  :      Optional[Any]					=   np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5			)
         lowercase  :      Optional[int]					=   np.concatenate([sigmas[::-1], [0.0]]			).astype(np.floataa			)
         lowercase  :      int					=   torch.from_numpy(lowerCAmelCase			)

         # setable values
         lowercase  :      List[str]					=   None
         lowercase  :      Tuple					=   np.linspace(0,     num_train_timesteps - 1,     lowerCAmelCase,     dtype=lowerCAmelCase			)[::-1].copy()
         lowercase  :      Dict					=   torch.from_numpy(lowerCAmelCase			)
         lowercase  :      str					=   False


   @property
   def        lowercase				(							self      :							Tuple			)    ->       Tuple:
         # standard deviation of the initial noise distribution
         if self.config.timestep_spacing in ["linspace", "trailing"]:
               return self.sigmas.max()

         return (self.sigmas.max() ** 2 + 1) ** 0.5


   def        lowercase				(							self      :							Any,     lowerCAmelCase      :							torch.FloatTensor,     lowerCAmelCase      :							Union[float, torch.FloatTensor]			)    ->       torch.FloatTensor:
         if isinstance(lowerCAmelCase,     torch.Tensor			):
               lowercase  :      Any					=   timestep.to(self.timesteps.device			)
         lowercase  :      List[Any]					=   (self.timesteps == timestep).nonzero().item()
         lowercase  :      Union[str, Any]					=   self.sigmas[step_index]
         lowercase  :      Dict					=   sample / ((sigma**2 + 1) ** 0.5)
         lowercase  :      Union[str, Any]					=   True
         return sample


   def        lowercase				(							self      :							Tuple,     lowerCAmelCase      :							int,     lowerCAmelCase      :							Union[str, torch.device] = None			)    ->       List[Any]:
         lowercase  :      Union[str, Any]					=   num_inference_steps

         # "linspace", "leading", "trailing" corresponds to annotation of Table 2. of https://arxiv.org/abs/2305.08891
         if self.config.timestep_spacing == "linspace":
               lowercase  :      int					=   np.linspace(0,     self.config.num_train_timesteps - 1,     lowerCAmelCase,     dtype=lowerCAmelCase			)[
                   ::-1
               ].copy()
         elif self.config.timestep_spacing == "leading":
               lowercase  :      List[str]					=   self.config.num_train_timesteps // self.num_inference_steps
               # creates integer timesteps by multiplying by ratio
               # casting to int to avoid issues when num_inference_step is power of 3
               lowercase  :      Any					=   (np.arange(0,     lowerCAmelCase			) * step_ratio).round()[::-1].copy().astype(lowerCAmelCase			)
               timesteps += self.config.steps_offset
         elif self.config.timestep_spacing == "trailing":
               lowercase  :      Optional[Any]					=   self.config.num_train_timesteps / self.num_inference_steps
               # creates integer timesteps by multiplying by ratio
               # casting to int to avoid issues when num_inference_step is power of 3
               lowercase  :      Optional[int]					=   (np.arange(self.config.num_train_timesteps,     0,     -step_ratio			)).round().copy().astype(lowerCAmelCase			)
               timesteps -= 1
         else:
               raise ValueError(
                   f'''{self.config.timestep_spacing} is not supported. Please make sure to choose one of \'linspace\', \'leading\' or \'trailing\'.'''			)

         lowercase  :      int					=   np.array(((1 - self.alphas_cumprod) / self.alphas_cumprod) ** 0.5			)
         lowercase  :      List[str]					=   np.interp(lowerCAmelCase,     np.arange(0,     len(lowerCAmelCase			)			),     lowerCAmelCase			)
         lowercase  :      Any					=   np.concatenate([sigmas, [0.0]]			).astype(np.floataa			)
         lowercase  :      Optional[Any]					=   torch.from_numpy(lowerCAmelCase			).to(device=lowerCAmelCase			)
         if str(lowerCAmelCase			).startswith('mps'			):
               # mps does not support float64
               lowercase  :      List[Any]					=   torch.from_numpy(lowerCAmelCase			).to(lowerCAmelCase,     dtype=torch.floataa			)
         else:
               lowercase  :      List[str]					=   torch.from_numpy(lowerCAmelCase			).to(device=lowerCAmelCase			)


   def        lowercase				(							self      :							Optional[Any],     lowerCAmelCase      :							torch.FloatTensor,     lowerCAmelCase      :							Union[float, torch.FloatTensor],     lowerCAmelCase      :							torch.FloatTensor,     lowerCAmelCase      :							Optional[torch.Generator] = None,     lowerCAmelCase      :							bool = True,     )    ->       Union[EulerAncestralDiscreteSchedulerOutput, Tuple]:

         if (
             isinstance(lowerCAmelCase,     lowerCAmelCase			)
             or isinstance(lowerCAmelCase,     torch.IntTensor			)
             or isinstance(lowerCAmelCase,     torch.LongTensor			)
         ):
               raise ValueError(
                   (
                       'Passing integer indices (e.g. from `enumerate(timesteps)`) as timesteps to'
                       ' `EulerDiscreteScheduler.step()` is not supported. Make sure to pass'
                       ' one of the `scheduler.timesteps` as a timestep.'
                   ),     )

         if not self.is_scale_input_called:
               logger.warning(
                   'The `scale_model_input` function should be called before `step` to ensure correct denoising. '
                   'See `StableDiffusionPipeline` for a usage example.'			)

         if isinstance(lowerCAmelCase,     torch.Tensor			):
               lowercase  :      List[str]					=   timestep.to(self.timesteps.device			)

         lowercase  :      List[str]					=   (self.timesteps == timestep).nonzero().item()
         lowercase  :      Optional[int]					=   self.sigmas[step_index]

         # 1. compute predicted original sample (x_0) from sigma-scaled predicted noise
         if self.config.prediction_type == "epsilon":
               lowercase  :      Optional[Any]					=   sample - sigma * model_output
         elif self.config.prediction_type == "v_prediction":
               # * c_out + input * c_skip
               lowercase  :      int					=   model_output * (-sigma / (sigma**2 + 1) ** 0.5) + (sample / (sigma**2 + 1))
         elif self.config.prediction_type == "sample":
               raise NotImplementedError('prediction_type not implemented yet: sample'			)
         else:
               raise ValueError(
                   f'''prediction_type given as {self.config.prediction_type} must be one of `epsilon`, or `v_prediction`'''			)

         lowercase  :      List[Any]					=   self.sigmas[step_index]
         lowercase  :      str					=   self.sigmas[step_index + 1]
         lowercase  :      Optional[int]					=   (sigma_to**2 * (sigma_from**2 - sigma_to**2) / sigma_from**2) ** 0.5
         lowercase  :      Optional[int]					=   (sigma_to**2 - sigma_up**2) ** 0.5

         # 2. Convert to an ODE derivative
         lowercase  :      Tuple					=   (sample - pred_original_sample) / sigma

         lowercase  :      str					=   sigma_down - sigma

         lowercase  :      int					=   sample + derivative * dt

         lowercase  :      Dict					=   model_output.device
         lowercase  :      List[Any]					=   randn_tensor(model_output.shape,     dtype=model_output.dtype,     device=lowerCAmelCase,     generator=lowerCAmelCase			)

         lowercase  :      List[str]					=   prev_sample + noise * sigma_up

         if not return_dict:
               return (prev_sample,)

         return EulerAncestralDiscreteSchedulerOutput(
             prev_sample=lowerCAmelCase,     pred_original_sample=lowerCAmelCase			)


   def        lowercase				(							self      :							int,     lowerCAmelCase      :							torch.FloatTensor,     lowerCAmelCase      :							torch.FloatTensor,     lowerCAmelCase      :							torch.FloatTensor,     )    ->       torch.FloatTensor:
         # Make sure sigmas and timesteps have the same device and dtype as original_samples
         lowercase  :      Tuple					=   self.sigmas.to(device=original_samples.device,     dtype=original_samples.dtype			)
         if original_samples.device.type == "mps" and torch.is_floating_point(lowerCAmelCase			):
               # mps does not support float64
               lowercase  :      Dict					=   self.timesteps.to(original_samples.device,     dtype=torch.floataa			)
               lowercase  :      Union[str, Any]					=   timesteps.to(original_samples.device,     dtype=torch.floataa			)
         else:
               lowercase  :      List[str]					=   self.timesteps.to(original_samples.device			)
               lowercase  :      Tuple					=   timesteps.to(original_samples.device			)

         lowercase  :      Union[str, Any]					=   [(schedule_timesteps == t).nonzero().item() for t in timesteps]

         lowercase  :      Union[str, Any]					=   sigmas[step_indices].flatten()
         while len(sigma.shape			) < len(original_samples.shape			):
               lowercase  :      Dict					=   sigma.unsqueeze(-1			)

         lowercase  :      Optional[Any]					=   original_samples + noise * sigma
         return noisy_samples






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