def bernoulli_number(m):
if m > 2 and m % 2 == 1:
return Rational(0)
out = 0
for k in range(m + 1):
for v in range(k + 1):
c = choose(k, v)
R = Rational(v**m, k + 1)
out += R * c * (-1)**v
return out
def bernstein_polynomial(v,n):
"""Bernstein Basis Polynomial"""
B = Polynomial([1,-1])
X = Polynomial([1])
bi = choose(n,v)
return X**v * B**(n-v) * bi
def binomial_eq(n, p, k):
return choose(n, k) * p**k * (1 - p)**(n - k)
from Rationals import bernoulli_number, Rational
from Polynomials import Polynomial
from Combinatorics import choose
for n in range(1, 9):
P = Polynomial([0])
for k in range(n + 1):
b = bernoulli_number(n - k)
c = choose(n, k)
co = b * c
P += Polynomial([0] * k + [co])
print(P)
print()
def neg_binomial_eq(r, p, k):
return choose(k + r - 1, k) * (1 - p)**r * p**k