




def lowerCamelCase__       (      _lowercase				):

     '''simple docstring'''


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

     UpperCAmelCase_      :							Tuple       =      [0] * (upper_limit + 1)

     # Base case: C(0) = C(1) = 1
     UpperCAmelCase_      :							List[Any]       =      1
     if upper_limit > 0:
          UpperCAmelCase_      :							List[Any]       =      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(_lowercase				):
               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:
               __a          =			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 __future__ import annotations

from fractions import Fraction







def lowerCamelCase__       (      _lowercase			,						_lowercase				):

     '''simple docstring'''


     return (
         num != den and num % 10 == den // 10 and (num // 10) / (den % 10) == num / den
     )







def lowerCamelCase__       (      _lowercase				):

     '''simple docstring'''


     UpperCAmelCase_      :							Optional[int]       =      []
     UpperCAmelCase_      :							Tuple       =      11
     UpperCAmelCase_      :							int       =      int('''1''' + '''0''' * digit_len				)
     for num in range(_lowercase			,						_lowercase				):
          while den <= 99:
               if (num != den) and (num % 10 == den // 10) and (den % 10 != 0):
                    if is_digit_cancelling(_lowercase			,						_lowercase				):
                         solutions.append(f'''{num}/{den}'''				)
               den += 1
          num += 1
          UpperCAmelCase_      :							Any       =      10
     return solutions







def lowerCamelCase__       (      _lowercase = 2				):

     '''simple docstring'''


     UpperCAmelCase_      :							Tuple       =      1.0
     for fraction in fraction_list(_lowercase				):
          UpperCAmelCase_      :							Optional[Any]       =      Fraction(_lowercase				)
          result *= frac.denominator / frac.numerator
     return int(_lowercase				)


if __name__ == "__main__":
     print(solution())