gistfile1.py

#https://math.stackexchange.com/questions/1189554/number-of-pairs-formed-from-2n-people-sitting-in-a-circle#1189713

import math, operator

product = lambda l: reduce(operator.mul, l, 1)

def binom(n,k): # https://stackoverflow.com/questions/26560726/python-binomial-coefficient
    if (k > n) or (k < 0):
        return 0
    return product(xrange(k+1, n+1)) // math.factorial(n-k)

def oddFactorial(n):  # computes 1*3*5*...*2n-1
    return product([2*k-1 for k in xrange(1,n+1)])

def Sam(n):
    return sum([(-1)**k * binom(2*n-k, k) * oddFactorial(n-k) for k in xrange(n+1)])


def Andrew(n):
    return sum([(-1)**k * (binom(2*n-k, k) + binom(2*n-k-1, k-1))*oddFactorial(n-k) for k in xrange(n+1)])

print [Sam(k) for k in xrange(2,10)]
print [Andrew(k) for k in xrange(2,10)]
print [Sam(k)- Sam(k-1) for k in xrange(2,10)]