
'''simple docstring'''
def    _UpperCamelCase						(       SCREAMING_SNAKE_CASE__ ,       SCREAMING_SNAKE_CASE__			)  ->							int:


		'''simple docstring'''


		snake_case  :       Optional[Any]     			=						1  # To kept the Calculated Value
		# Since C(n, k) = C(n, n-k)
		if k > (n - k):
				snake_case  :       Union[str, Any]     			=						n - k
		# Calculate C(n,k)
		for i in range(SCREAMING_SNAKE_CASE__			):
				result *= n - i
				result //= i + 1
		return result




def    _UpperCamelCase						(       SCREAMING_SNAKE_CASE__			)  ->							int:


		'''simple docstring'''


		return binomial_coefficient(2 * node_count ,       SCREAMING_SNAKE_CASE__			) // (node_count + 1)




def    _UpperCamelCase						(       SCREAMING_SNAKE_CASE__			)  ->							int:


		'''simple docstring'''


		if n < 0:
				raise ValueError('''factorial() not defined for negative values'''			)
		snake_case  :       Optional[int]     			=						1
		for i in range(1 ,       n + 1			):
				result *= i
		return result




def    _UpperCamelCase						(       SCREAMING_SNAKE_CASE__			)  ->							int:


		'''simple docstring'''


		return catalan_number(SCREAMING_SNAKE_CASE__			) * factorial(SCREAMING_SNAKE_CASE__			)


if __name__ == "__main__":
						lowercase__						       =      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."
						)










from __future__ import annotations






class 						lowercase_   :






      '''simple docstring'''




      def __init__( self :		List[str]					,   __UpperCAmelCase :		int    )       ->None:




       """simple docstring"""


       a						    = order

       # a_{0} ... a_{k}
       a						    = [1.0] + [0.0] * order
       # b_{0} ... b_{k}
       a						    = [1.0] + [0.0] * order

       # x[n-1] ... x[n-k]
       a						    = [0.0] * self.order
       # y[n-1] ... y[n-k]
       a						    = [0.0] * self.order







      def  __lowerCAmelCase   ( self :		Union[str, Any]					,   __UpperCAmelCase :		list[float]					,   __UpperCAmelCase :		list[float]    )       ->None:




       """simple docstring"""


       if len(__UpperCAmelCase    ) < self.order:
        a						    = [1.0, *a_coeffs]

       if len(__UpperCAmelCase    ) != self.order + 1:
        a						    = (
            F"""Expected a_coeffs to have {self.order + 1} elements """
            F"""for {self.order}-order filter, got {len(__UpperCAmelCase    )}"""
        )
        raise ValueError(__UpperCAmelCase    )

       if len(__UpperCAmelCase    ) != self.order + 1:
        a						    = (
            F"""Expected b_coeffs to have {self.order + 1} elements """
            F"""for {self.order}-order filter, got {len(__UpperCAmelCase    )}"""
        )
        raise ValueError(__UpperCAmelCase    )

       a						    = a_coeffs
       a						    = b_coeffs







      def  __lowerCAmelCase   ( self :		Any					,   __UpperCAmelCase :		float    )       ->float:




       """simple docstring"""


       a						    = 0.0

       # Start at index 1 and do index 0 at the end.
       for i in range(1					,   self.order + 1    ):
        result += (
            self.b_coeffs[i] * self.input_history[i - 1]
            - self.a_coeffs[i] * self.output_history[i - 1]
        )

       a						    = (result + self.b_coeffs[0] * sample) / self.a_coeffs[0]

       a						    = self.input_history[:-1]
       a						    = self.output_history[:-1]

       a						    = sample
       a						    = result

       return result


