import numpy as np
import math as m


def pascal(n: int) -> np.ndarray:
    p = np.zeros(n + 1)
    for i in range(n + 1):
        p[i] = m.factorial(n) / (m.factorial(i) * m.factorial(n - i))
    return p


def normalized_pascal(n: int) -> np.ndarray:
    return pascal(n) / 2 ** n


N = 10
heads = np.arange(N + 1)
print("heads:", heads)

print("pascal:", normalized_pascal(N))