



def 			lowerCAmelCase__	(					lowerCamelCase_	:	Any ,lowerCamelCase_	:	Optional[Any]):





				'''simple docstring'''

				lowerCAmelCase__  :       str				=  [0 for i in range(r + 1)]
				# nc0 = 1
				lowerCAmelCase__  :       Tuple				=  1
				for i in range(1 ,n + 1):
								# to compute current row from previous row.
								lowerCAmelCase__  :       Optional[int]				=  min(lowerCamelCase_ ,lowerCamelCase_)
								while j > 0:
												c[j] += c[j - 1]
												j -= 1
				return c[r]

print(binomial_coefficient(n=1_0, r=5))






def 			lowerCAmelCase__	(					lowerCamelCase_	:	int):





				'''simple docstring'''

				if length <= 0 or not isinstance(lowerCamelCase_ ,lowerCamelCase_):
								raise ValueError('''Length must be a positive integer.''')
				return [n * (2 * n - 1) for n in range(lowerCamelCase_)]


if __name__ == "__main__":
							print(hexagonal_numbers(length=5))
							print(hexagonal_numbers(length=1_0))

