
def 					lowerCamelCase__		(				_lowerCamelCase			,		_lowerCamelCase  )						->int:
 _UpperCAmelCase 				=1  # To kept the Calculated Value
 # Since C(n, k) = C(n, n-k)
 if k > (n - k):
  _UpperCAmelCase 				=n - k
 # Calculate C(n,k)
 for i in range(UpperCamelCase__  ):
  result *= n - i
  result //= i + 1
 return result
def 					lowerCamelCase__		(				_lowerCamelCase  )						->int:
 return binomial_coefficient(2 * node_count			,		UpperCamelCase__  ) // (node_count + 1)
def 					lowerCamelCase__		(				_lowerCamelCase  )						->int:
 if n < 0:
  raise ValueError("factorial() not defined for negative values"  )
 _UpperCAmelCase 				=1
 for i in range(1			,		n + 1  ):
  result *= i
 return result
def 					lowerCamelCase__		(				_lowerCamelCase  )						->int:
 return catalan_number(UpperCamelCase__  ) * factorial(UpperCamelCase__  )


if __name__ == "__main__":
       snake_case__      :		Optional[int]				=       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."""
       )












import itertools
import math


def       UpperCamelCase__(			UpperCamelCase__     :			int    )->bool:

       if 1 < number < 4:
              # 2 and 3 are primes
              return True
       elif number < 2 or number % 2 == 0 or number % 3 == 0:
              # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
              return False

       # All primes number are in format of 6k +/- 1
       for i in range(5					,	int(math.sqrt(UpperCamelCase__    ) + 1    )					,	6    ):
              if number % i == 0 or number % (i + 2) == 0:
                     return False
       return True


def       UpperCamelCase__(			)->Tuple:
       A__		   =		2
       while True:
              if is_prime(UpperCamelCase__    ):
                     yield num
              num += 1


def       UpperCamelCase__(			UpperCamelCase__     :			int = 1_00_01    )->int:
       return next(itertools.islice(prime_generator()					,	nth - 1					,	UpperCamelCase__    )    )


if __name__ == "__main__":
       print(F"{solution() = }")





