如何用非整数计算二项式系数

Question

我知道如何计算select（5,2）的二项式系数，但是现在我想知道是否存在可以用python或R编程语言计算select（5,2.1）的函数？

Answer 1

The combination formula for "n choose k" is given by

where n and k are whole numbers. The generalized version for x and y in the set of real numbers is given by

where Γ(x) is the gamma function, a generalized form of the factorial.

To create this in Python, you can use the following:

import math

def generalized_binomial(x,y):
    return math.gamma(x+1) / (math.gamma(y+1) * math.gamma(x-y+1))

generalized_binomial(5,2.1)
# returns:
10.304042688575835

Answer 2

Use the gamma function from scipy and the extended definition of the binomial coefficient.

>>> from scipy.special import gamma
>>> def choose(x,y):
...     return gamma(x+1)/(gamma(y+1)*gamma(x-y+1))
... 
>>> choose(5,2.1)
10.304042688575837

Answer 3

In R you could make a function like this using gamma :

choose <- function(x, y) {
  return(gamma(x+1)/(gamma(y+1)*gamma(x-y+1)))
}

print(choose(5,2.1)) # 10.30404