





'''simple docstring'''

def  a__    (    lowercase :					int				)		->		"list[int]":





     """simple docstring"""

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

     _UpperCamelCase       =	[0] * (upper_limit + 1)

     # Base case: C(0) = C(1) = 1
     _UpperCamelCase       =	1
     if upper_limit > 0:
          _UpperCamelCase       =	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(_lowerCAmelCase				):
               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:
         lowercase__						:      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()










'''simple docstring'''

def  a__    (    lowercase :					int				)		->		int:





     """simple docstring"""

     if not isinstance(lowercase, lowercase				):
          raise TypeError('''Input value must be an \'int\' type'''				)

     _UpperCamelCase       =	0
     while number:
          position += 1
          number >>= 1

     return position


if __name__ == "__main__":
   import doctest

   doctest.testmod()



