import numpy as np
from scipy.special import binom

def a(n):
    if n == 0:
        return 1
    sum = 0
    for k in range(1, n+1):
        sum += np.power(-1, k-1) * binom(n, k) * a(n-k)*np.power(2,k*(n-k))
    return sum

def a_prime(n):
    if n == 0 or n == 1:
        return 1
    sum = 0
    for k in range(1, n+1):
        sum += np.power(-1, k-1) * binom(n, k) * a_prime(n-k)*np.power(np.power(2, n-k) - 1, k)
    return sum

def a_prime_prime(n):
    if n == 0 or n == 1:
        return 1
    sum = 0
    for k in range(1, n):
        summand = np.power(-1, k-1) * binom(n, k) * k * a_prime(n-k)*np.power(np.power(2, n-k) - 1, k)
        print(summand)
        sum += summand
    return sum