






def      lowerCAmelCase__	(    _a       :			int  ,      _a       :			int    ):
		snake_case_				:							Optional[int]						 =       1  # To kept the Calculated Value
		# Since C(n, k) = C(n, n-k)
		if k > (n - k):
				snake_case_				:							int						 =       n - k
		# Calculate C(n,k)
		for i in range(_a    ):
				result *= n - i
				result //= i + 1
		return result

def      lowerCAmelCase__	(    _a       :			int    ):
		return binomial_coefficient(2 * node_count  ,      _a    ) // (node_count + 1)

def      lowerCAmelCase__	(    _a       :			int    ):
		if n < 0:
				raise ValueError("factorial() not defined for negative values"    )
		snake_case_				:							Union[str, Any]						 =       1
		for i in range(1  ,      n + 1    ):
				result *= i
		return result

def      lowerCAmelCase__	(    _a       :			int    ):
		return catalan_number(_a    ) * factorial(_a    )


if __name__ == "__main__":
							lowercase  :							Any			  =    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."""
							)








def      lowerCAmelCase__	(    _a       :			str  ,      _a       :			int    ):
		# "extended trapezoidal rule"
		# int(f) = dx/2 * (f1 + 2f2 + ... + fn)
		snake_case_				:							Optional[Any]						 =       (boundary[1] - boundary[0]) / steps
		snake_case_				:							str						 =       boundary[0]
		snake_case_				:							Dict						 =       boundary[1]
		snake_case_				:							str						 =       make_points(_a  ,      _a  ,      _a    )
		snake_case_				:							str						 =       0.0
		y += (h / 2.0) * f(_a    )
		for i in x_i:
				# print(i)
				y += h * f(_a    )
		y += (h / 2.0) * f(_a    )
		return y

def      lowerCAmelCase__	(    _a       :			str  ,      _a       :			Optional[Any]  ,      _a       :			Tuple    ):
		snake_case_				:							Tuple						 =       a + h
		while x < (b - h):
				yield x
				snake_case_				:							List[str]						 =       x + h

def      lowerCAmelCase__	(    _a       :			Tuple    ):  # enter your function here
		snake_case_				:							Tuple						 =       (x - 0) * (x - 0)
		return y

def      lowerCAmelCase__	(    ):
		snake_case_				:							Union[str, Any]						 =       0.0  # Lower bound of integration
		snake_case_				:							List[Any]						 =       1.0  # Upper bound of integration
		snake_case_				:							Tuple						 =       10.0  # define number of steps or resolution
		snake_case_				:							List[Any]						 =       [a, b]  # define boundary of integration
		snake_case_				:							Any						 =       method_a(_a  ,      _a    )
		print(F'''y = {y}'''    )


if __name__ == "__main__":
							main()
