




from __future__ import annotations



def      __lowerCAmelCase			(					UpperCamelCase      )     ->	list[int]:  # This function is recursive
  lowerCAmelCase__				:  Tuple		       =     len(UpperCamelCase      )
  # If the array contains only one element, we return it (it's the stop condition of
  # recursion)
  if array_length <= 1:
    return array
    # Else
  lowerCAmelCase__				:  Optional[int]		       =     array[0]
  lowerCAmelCase__				:  Union[str, Any]		       =     False
  lowerCAmelCase__				:  List[str]		       =     1
  lowerCAmelCase__				:  list[int]		       =     []
  while not is_found and i < array_length:
    if array[i] < pivot:
      lowerCAmelCase__				:  List[Any]		       =     True
      lowerCAmelCase__				:  Optional[int]		       =     [element for element in array[i:] if element >= array[i]]
      lowerCAmelCase__				:  Any		       =     longest_subsequence(UpperCamelCase      )
      if len(UpperCamelCase      ) > len(UpperCamelCase      ):
        lowerCAmelCase__				:  Optional[int]		       =     temp_array
    else:
      i += 1

  lowerCAmelCase__				:  Optional[Any]		       =     [element for element in array[1:] if element >= pivot]
  lowerCAmelCase__				:  List[Any]		       =     [pivot, *longest_subsequence(UpperCamelCase      )]
  if len(UpperCamelCase      ) > len(UpperCamelCase      ):
    return temp_array
  else:
    return longest_subseq


if __name__ == "__main__":
     import doctest

     doctest.testmod()







def      __lowerCAmelCase			(					UpperCamelCase      )     ->	None:
  lowerCAmelCase__				:  Dict		       =     generate_pascal_triangle(UpperCamelCase      )
  for row_idx in range(UpperCamelCase      ):
    # Print left spaces
    for _ in range(num_rows - row_idx - 1      ):
      print(end=''' '''      )
    # Print row values
    for col_idx in range(row_idx + 1      ):
      if col_idx != row_idx:
        print(triangle[row_idx][col_idx]	,      end=''' '''      )
      else:
        print(triangle[row_idx][col_idx]	,      end=''''''      )
    print()



def      __lowerCAmelCase			(					UpperCamelCase      )     ->	list[list[int]]:

  if not isinstance(UpperCamelCase	,      UpperCamelCase      ):
    raise TypeError('''The input value of \'num_rows\' should be \'int\''''      )

  if num_rows == 0:
    return []
  elif num_rows < 0:
    raise ValueError(
        '''The input value of \'num_rows\' should be greater than or equal to 0'''      )

  lowerCAmelCase__				:  list[list[int]]		       =     []
  for current_row_idx in range(UpperCamelCase      ):
    lowerCAmelCase__				:  List[str]		       =     populate_current_row(UpperCamelCase	,      UpperCamelCase      )
    triangle.append(UpperCamelCase      )
  return triangle



def      __lowerCAmelCase			(					UpperCamelCase	,      UpperCamelCase      )     ->	list[int]:
  lowerCAmelCase__				:  Optional[Any]		       =     [-1] * (current_row_idx + 1)
  # first and last elements of current row are equal to 1
  lowerCAmelCase__       ,			lowerCAmelCase__				:  int		       =     1, 1
  for current_col_idx in range(1	,      UpperCamelCase      ):
    calculate_current_element(
        UpperCamelCase	,      UpperCamelCase	,      UpperCamelCase	,      UpperCamelCase      )
  return current_row



def      __lowerCAmelCase			(					UpperCamelCase	,      UpperCamelCase	,      UpperCamelCase	,      UpperCamelCase	,      )     ->	None:
  lowerCAmelCase__				:  Optional[int]		       =     triangle[current_row_idx - 1][current_col_idx - 1]
  lowerCAmelCase__				:  List[str]		       =     triangle[current_row_idx - 1][current_col_idx]
  lowerCAmelCase__				:  List[Any]		       =     above_to_left_elt + above_to_right_elt



def      __lowerCAmelCase			(					UpperCamelCase      )     ->	list[list[int]]:

  if not isinstance(UpperCamelCase	,      UpperCamelCase      ):
    raise TypeError('''The input value of \'num_rows\' should be \'int\''''      )

  if num_rows == 0:
    return []
  elif num_rows < 0:
    raise ValueError(
        '''The input value of \'num_rows\' should be greater than or equal to 0'''      )

  lowerCAmelCase__				:  list[list[int]]		       =     [[1]]

  for row_index in range(1	,      UpperCamelCase      ):
    lowerCAmelCase__				:  Optional[int]		       =     [0] + result[-1] + [0]
    lowerCAmelCase__				:  List[str]		       =     row_index + 1
    # Calculate the number of distinct elements in a row
    lowerCAmelCase__				:  Tuple		       =     sum(divmod(UpperCamelCase	,      2      )      )
    lowerCAmelCase__				:  Tuple		       =     [
        temp_row[i - 1] + temp_row[i] for i in range(1	,      distinct_elements + 1      )
    ]
    lowerCAmelCase__				:  int		       =     row_first_half[: (row_index + 1) // 2]
    row_second_half.reverse()
    lowerCAmelCase__				:  Dict		       =     row_first_half + row_second_half
    result.append(UpperCamelCase      )

  return result



def      __lowerCAmelCase			(					)     ->	None:
  from collections.abc import Callable
  from timeit import timeit

  def benchmark_a_function(UpperCamelCase	,      UpperCamelCase      ) -> None:
    lowerCAmelCase__				:  Tuple		       =     F"""{func.__name__}({value})"""
    lowerCAmelCase__				:  Dict		       =     timeit(F"""__main__.{call}"""	,      setup='''import __main__'''      )
    # print(f"{call:38} = {func(value)} -- {timing:.4f} seconds")
    print(F"""{call:38} -- {timing:.4f} seconds"""      )

  for value in range(15      ):  # (1, 7, 14):
    for func in (generate_pascal_triangle, generate_pascal_triangle_optimized):
      benchmark_a_function(UpperCamelCase	,      UpperCamelCase      )
    print()


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     benchmark()

