





def lowerCAmelCase__(__snake_case							)				->      Union[str, Any]:







	'''simple docstring'''







	lowerCamelCase__								=		generate_pascal_triangle(__snake_case							)
	for row_idx in range(__snake_case							):
		# Print left spaces
		for _ in range(num_rows - row_idx - 1							):
			print(end=''' '''							)
		# Print row values
		for col_idx in range(row_idx + 1							):
			if col_idx != row_idx:
				print(triangle[row_idx][col_idx]       ,end=''' '''							)
			else:
				print(triangle[row_idx][col_idx]       ,end=''''''							)
		print()





def lowerCAmelCase__(__snake_case							)				->      List[Any]:







	'''simple docstring'''







	if not isinstance(__snake_case       ,__snake_case							):
		raise TypeError('''The input value of \'num_rows\' should be \'int\''''							)

	if num_rows == 0:
		return []
	elif num_rows < 0:
		raise ValueError(
		    '''The input value of \'num_rows\' should be greater than or equal to 0'''							)

	lowerCamelCase__								=		[]
	for current_row_idx in range(__snake_case							):
		lowerCamelCase__								=		populate_current_row(__snake_case       ,__snake_case							)
		triangle.append(__snake_case							)
	return triangle





def lowerCAmelCase__(__snake_case       ,__snake_case							)				->      List[str]:







	'''simple docstring'''







	lowerCamelCase__								=		[-1] * (current_row_idx + 1)
	# first and last elements of current row are equal to 1
	lowerCamelCase__				,	lowerCamelCase__								=		1, 1
	for current_col_idx in range(1       ,__snake_case							):
		calculate_current_element(
		    __snake_case       ,__snake_case       ,__snake_case       ,__snake_case							)
	return current_row





def lowerCAmelCase__(__snake_case       ,__snake_case       ,__snake_case       ,__snake_case       ,)				->      int:







	'''simple docstring'''







	lowerCamelCase__								=		triangle[current_row_idx - 1][current_col_idx - 1]
	lowerCamelCase__								=		triangle[current_row_idx - 1][current_col_idx]
	lowerCamelCase__								=		above_to_left_elt + above_to_right_elt





def lowerCAmelCase__(__snake_case							)				->      Optional[Any]:







	'''simple docstring'''







	if not isinstance(__snake_case       ,__snake_case							):
		raise TypeError('''The input value of \'num_rows\' should be \'int\''''							)

	if num_rows == 0:
		return []
	elif num_rows < 0:
		raise ValueError(
		    '''The input value of \'num_rows\' should be greater than or equal to 0'''							)

	lowerCamelCase__								=		[[1]]

	for row_index in range(1       ,__snake_case							):
		lowerCamelCase__								=		[0] + result[-1] + [0]
		lowerCamelCase__								=		row_index + 1
		# Calculate the number of distinct elements in a row
		lowerCamelCase__								=		sum(divmod(__snake_case       ,2							)							)
		lowerCamelCase__								=		[
		    temp_row[i - 1] + temp_row[i] for i in range(1       ,distinct_elements + 1							)
		]
		lowerCamelCase__								=		row_first_half[: (row_index + 1) // 2]
		row_second_half.reverse()
		lowerCamelCase__								=		row_first_half + row_second_half
		result.append(__snake_case							)

	return result





def lowerCAmelCase__()				->      Optional[Any]:







	'''simple docstring'''







	from collections.abc import Callable
	from timeit import timeit

	def benchmark_a_function(__snake_case       ,__snake_case							) -> None:
		lowerCamelCase__								=		F'{func.__name__}({value})'
		lowerCamelCase__								=		timeit(F'__main__.{call}'       ,setup='''import __main__'''							)
		# print(f"{call:38} = {func(value)} -- {timing:.4f} seconds")
		print(F'{call:38} -- {timing:.4f} seconds'							)

	for value in range(15							):  # (1, 7, 14):
		for func in (generate_pascal_triangle, generate_pascal_triangle_optimized):
			benchmark_a_function(__snake_case       ,__snake_case							)
		print()


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			benchmark()







from __future__ import annotations
def 			_a (      SCREAMING_SNAKE_CASE						,				SCREAMING_SNAKE_CASE     ):


		"""simple docstring"""




		lowercase__            =      0
		lowercase__            =      len(SCREAMING_SNAKE_CASE     ) - 1

		while i < j:
				if nums[i] + nums[j] == target:
						return [i, j]
				elif nums[i] + nums[j] < target:
						lowercase__            =      i + 1
				else:
						lowercase__            =      j - 1

		return []


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			print(f"""{two_pointer([2, 7, 11, 15], 9) = }""")


