Buggy python submission (verdict: Runtime Error):
import numpy as np
from collections import defaultdict

class Combinatorics:
    def __init__(self, N, mod):
        '''
        Preprocess for calculating binomial coefficients nCr (0 <= r <= n, 0 <= n <= N)
        over the finite field Z/(mod)Z.
        Input:
            N (int): maximum n
            mod (int): a prime number. The order of the field Z/(mod)Z over which nCr is calculated.
        '''
        self.mod = mod
        self.fact = {i: None for i in range(N+1)}         # n!
        self.inverse = {i: None for i in range(1, N+1)}   # inverse of n in the field Z/(MOD)Z
        self.fact_inverse = {i: None for i in range(N+1)} # inverse of n! in the field Z/(MOD)Z
        
        # preprocess
        self.fact[0] = self.fact[1] = 1
        self.fact_inverse[0] = self.fact_inverse[1] = 1
        self.inverse[1] = 1
        for i in range(2, N+1):
            self.fact[i] = i * self.fact[i-1] % self.mod
            q, r = divmod(self.mod, i)
            self.inverse[i] = (- (q % self.mod) * self.inverse[r]) % self.mod
            self.fact_inverse[i] = self.inverse[i] * self.fact_inverse[i-1] % self.mod
    
    def binom(self, n, r):
        '''
        Calculate nCr = n! /(r! (n-r)!) % mod
        '''
        if n < r or n < 0 or r < 0:
            return 0
        else:
            return self.fact[n] * (self.fact_inverse[r] * self.fact_inverse[n-r] % self.mod) % self.mod

N, M = map(int, input().split())
com = Combinatorics(N, M)
ans = 0

# Preprocess
# calculate 2**n and 2**(2**n)
pow2 = np.power(2, np.array(range(N*N // 4 + 1)))
pow_pow2 = [0] * (N+1)
pow_pow2[0] = 2
for i in range(1, N*N // 4 + 1):
    pow2[i] = (pow2[i-1] * 2) % M
for i in range(1, N+1):
    pow_pow2[i] = pow(pow_pow2[i-1], 2, M)

# (#ways to order i ramens, with each of n toppings chosen at most once)
ways2 = {n: defaultdict(int) for n in range(N+1)}
for n in range(N+1):
    ways = 0 # (#ways to choose n toppings, with each topping chosen at most once)
    temp = 0
    if n >= 1:
        for i in range(n+1):
            ways2[n][i] = (ways2[n][i] +  ways2[n-1][i]) % M # n-th topping is not used
            if i >= 1: ways2[n][i] = (ways2[n][i] + ways2[n-1][i-1]) % M # only n-th topping is used in an order
            ways2[n][i] = (ways2[n][i] + (i*ways2[n-1][i]) % M) % M # n-th topping is chosen with another topping
            temp = (temp + (ways2[n][i] * pow2[(N-n)*i]) % M) % M # degree of freedom on each order is 2**(N-n)
    else: # n = 0
        ways2[n][0] = 1
        temp += ways2[n][0]
    ways = (temp * pow_pow2[N-n]) % M
    if n % 2 == 0:
        ans = (ans + (com.binom(N, n) * ways) % M) % M
    else:
        ans = (ans - (com.binom(N, n) * ways) % M) % M
print(ans)

Fixed submission:
from collections import defaultdict

class Combinatorics:
    def __init__(self, N, mod):
        '''
        Preprocess for calculating binomial coefficients nCr (0 <= r <= n, 0 <= n <= N)
        over the finite field Z/(mod)Z.
        Input:
            N (int): maximum n
            mod (int): a prime number. The order of the field Z/(mod)Z over which nCr is calculated.
        '''
        self.mod = mod
        self.fact = {i: None for i in range(N+1)}         # n!
        self.inverse = {i: None for i in range(1, N+1)}   # inverse of n in the field Z/(MOD)Z
        self.fact_inverse = {i: None for i in range(N+1)} # inverse of n! in the field Z/(MOD)Z
        
        # preprocess
        self.fact[0] = self.fact[1] = 1
        self.fact_inverse[0] = self.fact_inverse[1] = 1
        self.inverse[1] = 1
        for i in range(2, N+1):
            self.fact[i] = i * self.fact[i-1] % self.mod
            q, r = divmod(self.mod, i)
            self.inverse[i] = (- (q % self.mod) * self.inverse[r]) % self.mod
            self.fact_inverse[i] = self.inverse[i] * self.fact_inverse[i-1] % self.mod
    
    def binom(self, n, r):
        '''
        Calculate nCr = n! /(r! (n-r)!) % mod
        '''
        if n < r or n < 0 or r < 0:
            return 0
        else:
            return self.fact[n] * (self.fact_inverse[r] * self.fact_inverse[n-r] % self.mod) % self.mod

N, M = map(int, input().split())
com = Combinatorics(N, M)
ans = 0

# Preprocess
# calculate 2**n and 2**(2**n)
pow2 = [0] * (N*N // 4 + 1)
pow_pow2 = [0] * (N+1)
pow2[0] = 1; pow_pow2[0] = 2
for i in range(1, N*N // 4 + 1):
    pow2[i] = (pow2[i-1] * 2) % M
for i in range(1, N+1):
    pow_pow2[i] = pow(pow_pow2[i-1], 2, M)

# (#ways to order i ramens, with each of n toppings chosen at most once)
ways2 = {n: defaultdict(int) for n in range(N+1)}
for n in range(N+1):
    ways = 0 # (#ways to choose n toppings, with each topping chosen at most once)
    temp = 0
    if n >= 1:
        for i in range(n+1):
            ways2[n][i] = (ways2[n][i] +  ways2[n-1][i]) % M # n-th topping is not used
            if i >= 1: ways2[n][i] = (ways2[n][i] + ways2[n-1][i-1]) % M # only n-th topping is used in an order
            ways2[n][i] = (ways2[n][i] + (i*ways2[n-1][i]) % M) % M # n-th topping is chosen with another topping
            temp = (temp + (ways2[n][i] * pow2[(N-n)*i]) % M) % M # degree of freedom on each order is 2**(N-n)
    else: # n = 0
        ways2[n][0] = 1
        temp += ways2[n][0]
    ways = (temp * pow_pow2[N-n]) % M
    if n % 2 == 0:
        ans = (ans + (com.binom(N, n) * ways) % M) % M
    else:
        ans = (ans - (com.binom(N, n) * ways) % M) % M
print(ans)

Buggy python submission (verdict: Runtime Error):
class Combinatorics:
    def __init__(self, N, mod):
        '''
        Preprocess for calculating binomial coefficients nCr (0 <= r <= n, 0 <= n <= N)
        over the finite field Z/(mod)Z.
        Input:
            N (int): maximum n
            mod (int): a prime number. The order of the field Z/(mod)Z over which nCr is calculated.
        '''
        self.mod = mod
        self.fact = {i: None for i in range(N+1)}         # n!
        self.inverse = {i: None for i in range(1, N+1)}   # inverse of n in the field Z/(MOD)Z
        self.fact_inverse = {i: None for i in range(N+1)} # inverse of n! in the field Z/(MOD)Z
        
        # preprocess
        self.fact[0] = self.fact[1] = 1
        self.fact_inverse[0] = self.fact_inverse[1] = 1
        self.inverse[1] = 1
        for i in range(2, N+1):
            self.fact[i] = i * self.fact[i-1] % self.mod
            q, r = divmod(self.mod, i)
            self.inverse[i] = (- (q % self.mod) * self.inverse[r]) % self.mod
            self.fact_inverse[i] = self.inverse[i] * self.fact_inverse[i-1] % self.mod
    
    def binom(self, n, r):
        '''
        Calculate nCr = n! /(r! (n-r)!) % mod
        '''
        if n < r or n < 0 or r < 0:
            return 0
        else:
            return self.fact[n] * (self.fact_inverse[r] * self.fact_inverse[n-r] % self.mod) % self.mod

N, M = map(int, input().split())
com = Combinatorics(N, M)
ans = 0

# Preprocess
# calculate 2**n and 2**(2**n)
pow2 = [0] * (N*N // 4 + 1)
pow_pow2 = [0] * (N+1)
pow2[0] = 1; pow_pow2[0] = 2
for i in range(1, N*N + 1):
    pow2[i] = (pow2[i-1] * 2) % M
for i in range(1, N+1):
    pow_pow2[i] = pow(pow_pow2[i-1], 2, M)

# (#ways to order i ramens, with each of n toppings chosen at most once)
ways2 = {n: {i: 0 for i in range(N+1)} for n in range(N+1)}
for n in range(N+1):
    ways = 0 # (#ways to choose n toppings, with each topping chosen at most once)
    temp = 0
    if n >= 1:
        for i in range(n+1):
            ways2[n][i] = (ways2[n][i] +  ways2[n-1][i]) % M # n-th topping is not used
            if i >= 1: ways2[n][i] = (ways2[n][i] + ways2[n-1][i-1]) % M # only n-th topping is used in an order
            ways2[n][i] = (ways2[n][i] + (i*ways2[n-1][i]) % M) % M # n-th topping is chosen with another topping
            temp = (temp + (ways2[n][i] * pow2[(N-n)*i]) % M) % M # degree of freedom on each order is 2**(N-n)
    else: # n = 0
        ways2[n][0] = 1
        temp += ways2[n][0]
    ways = (temp * pow_pow2[N-n]) % M
    if n % 2 == 0:
        ans = (ans + (com.binom(N, n) * ways) % M) % M
    else:
        ans = (ans - (com.binom(N, n) * ways) % M) % M
print(ans)


Fixed submission:
from collections import defaultdict

class Combinatorics:
    def __init__(self, N, mod):
        '''
        Preprocess for calculating binomial coefficients nCr (0 <= r <= n, 0 <= n <= N)
        over the finite field Z/(mod)Z.
        Input:
            N (int): maximum n
            mod (int): a prime number. The order of the field Z/(mod)Z over which nCr is calculated.
        '''
        self.mod = mod
        self.fact = {i: None for i in range(N+1)}         # n!
        self.inverse = {i: None for i in range(1, N+1)}   # inverse of n in the field Z/(MOD)Z
        self.fact_inverse = {i: None for i in range(N+1)} # inverse of n! in the field Z/(MOD)Z
        
        # preprocess
        self.fact[0] = self.fact[1] = 1
        self.fact_inverse[0] = self.fact_inverse[1] = 1
        self.inverse[1] = 1
        for i in range(2, N+1):
            self.fact[i] = i * self.fact[i-1] % self.mod
            q, r = divmod(self.mod, i)
            self.inverse[i] = (- (q % self.mod) * self.inverse[r]) % self.mod
            self.fact_inverse[i] = self.inverse[i] * self.fact_inverse[i-1] % self.mod
    
    def binom(self, n, r):
        '''
        Calculate nCr = n! /(r! (n-r)!) % mod
        '''
        if n < r or n < 0 or r < 0:
            return 0
        else:
            return self.fact[n] * (self.fact_inverse[r] * self.fact_inverse[n-r] % self.mod) % self.mod

N, M = map(int, input().split())
com = Combinatorics(N, M)
ans = 0

# Preprocess
# calculate 2**n and 2**(2**n)
pow2 = [0] * (N*N // 4 + 1)
pow_pow2 = [0] * (N+1)
pow2[0] = 1; pow_pow2[0] = 2
for i in range(1, N*N // 4 + 1):
    pow2[i] = (pow2[i-1] * 2) % M
for i in range(1, N+1):
    pow_pow2[i] = pow(pow_pow2[i-1], 2, M)

# (#ways to order i ramens, with each of n toppings chosen at most once)
ways2 = {n: defaultdict(int) for n in range(N+1)}
for n in range(N+1):
    ways = 0 # (#ways to choose n toppings, with each topping chosen at most once)
    temp = 0
    if n >= 1:
        for i in range(n+1):
            ways2[n][i] = (ways2[n][i] +  ways2[n-1][i]) % M # n-th topping is not used
            if i >= 1: ways2[n][i] = (ways2[n][i] + ways2[n-1][i-1]) % M # only n-th topping is used in an order
            ways2[n][i] = (ways2[n][i] + (i*ways2[n-1][i]) % M) % M # n-th topping is chosen with another topping
            temp = (temp + (ways2[n][i] * pow2[(N-n)*i]) % M) % M # degree of freedom on each order is 2**(N-n)
    else: # n = 0
        ways2[n][0] = 1
        temp += ways2[n][0]
    ways = (temp * pow_pow2[N-n]) % M
    if n % 2 == 0:
        ans = (ans + (com.binom(N, n) * ways) % M) % M
    else:
        ans = (ans - (com.binom(N, n) * ways) % M) % M
print(ans)

Buggy python submission (verdict: Time Limit Exceeded):

N, M = map(int, input().split())
fact = {i: None for i in range(N+1)}
fact_inverse = {i: None for i in range(N+1)}

fact[0] = fact[1] = 1
fact_inverse[0] = fact_inverse[1] = 1
for i in range(2,N+1):
    fact[i] = (fact[i-1]*i) % M
    fact_inverse[i] = pow(fact[i], M-2, M)
    
def binom(n, r):
    '''
    Calculate nCr = n! /(r! (n-r)!) % mod
    '''
    if n < r or n < 0 or r < 0:
        return 0
    else:
        return fact[n] * (fact_inverse[r] * fact_inverse[n-r] % M) % M

ans = 0

# Preprocess
# calculate 2**(2**n)
pow_pow2 = [0] * (N+1)
pow_pow2[0] = 2
for i in range(1, N+1):
    pow_pow2[i] = pow(pow_pow2[i-1], 2, M)

# (#ways to order i ramens, with each of n toppings chosen at most once)
ways2 = {n: {i: 0 for i in range(N+1)} for n in range(N+1)}
for n in range(N+1):
    ways = 0 # (#ways to choose n toppings, with each topping chosen at most once)
    temp = 0
    if n >= 1:
        for i in range(n+1):
            ways2[n][i] = (ways2[n][i] +  ways2[n-1][i]) % M # n-th topping is not used
            if i >= 1: ways2[n][i] = (ways2[n][i] + ways2[n-1][i-1]) % M # only n-th topping is used in an order
            ways2[n][i] = (ways2[n][i] + (i*ways2[n-1][i]) % M) % M # n-th topping is chosen with another topping
            temp = (temp + (ways2[n][i] * pow(2, (N-n)*i, M)) % M) % M # degree of freedom on each order is 2**(N-n)
    else: # n = 0
        ways2[n][0] = 1
        temp += ways2[n][0]
    ways = (temp * pow_pow2[N-n]) % M
    if n % 2 == 0:
        ans = (ans + (binom(N, n) * ways) % M) % M
    else:
        ans = (ans - (binom(N, n) * ways) % M) % M
print(ans)

Fixed submission:
from collections import defaultdict

class Combinatorics:
    def __init__(self, N, mod):
        '''
        Preprocess for calculating binomial coefficients nCr (0 <= r <= n, 0 <= n <= N)
        over the finite field Z/(mod)Z.
        Input:
            N (int): maximum n
            mod (int): a prime number. The order of the field Z/(mod)Z over which nCr is calculated.
        '''
        self.mod = mod
        self.fact = {i: None for i in range(N+1)}         # n!
        self.inverse = {i: None for i in range(1, N+1)}   # inverse of n in the field Z/(MOD)Z
        self.fact_inverse = {i: None for i in range(N+1)} # inverse of n! in the field Z/(MOD)Z
        
        # preprocess
        self.fact[0] = self.fact[1] = 1
        self.fact_inverse[0] = self.fact_inverse[1] = 1
        self.inverse[1] = 1
        for i in range(2, N+1):
            self.fact[i] = i * self.fact[i-1] % self.mod
            q, r = divmod(self.mod, i)
            self.inverse[i] = (- (q % self.mod) * self.inverse[r]) % self.mod
            self.fact_inverse[i] = self.inverse[i] * self.fact_inverse[i-1] % self.mod
    
    def binom(self, n, r):
        '''
        Calculate nCr = n! /(r! (n-r)!) % mod
        '''
        if n < r or n < 0 or r < 0:
            return 0
        else:
            return self.fact[n] * (self.fact_inverse[r] * self.fact_inverse[n-r] % self.mod) % self.mod

N, M = map(int, input().split())
com = Combinatorics(N, M)
ans = 0

# Preprocess
# calculate 2**n and 2**(2**n)
pow2 = [0] * (N*N // 4 + 1)
pow_pow2 = [0] * (N+1)
pow2[0] = 1; pow_pow2[0] = 2
for i in range(1, N*N // 4 + 1):
    pow2[i] = (pow2[i-1] * 2) % M
for i in range(1, N+1):
    pow_pow2[i] = pow(pow_pow2[i-1], 2, M)

# (#ways to order i ramens, with each of n toppings chosen at most once)
ways2 = {n: defaultdict(int) for n in range(N+1)}
for n in range(N+1):
    ways = 0 # (#ways to choose n toppings, with each topping chosen at most once)
    temp = 0
    if n >= 1:
        for i in range(n+1):
            ways2[n][i] = (ways2[n][i] +  ways2[n-1][i]) % M # n-th topping is not used
            if i >= 1: ways2[n][i] = (ways2[n][i] + ways2[n-1][i-1]) % M # only n-th topping is used in an order
            ways2[n][i] = (ways2[n][i] + (i*ways2[n-1][i]) % M) % M # n-th topping is chosen with another topping
            temp = (temp + (ways2[n][i] * pow2[(N-n)*i]) % M) % M # degree of freedom on each order is 2**(N-n)
    else: # n = 0
        ways2[n][0] = 1
        temp += ways2[n][0]
    ways = (temp * pow_pow2[N-n]) % M
    if n % 2 == 0:
        ans = (ans + (com.binom(N, n) * ways) % M) % M
    else:
        ans = (ans - (com.binom(N, n) * ways) % M) % M
print(ans)

Buggy python submission (verdict: Runtime Error):

N, M = map(int, input().split())
fact = {i: None for i in range(N+1)}
fact_inverse = {i: None for i in range(N+1)}

fact[0] = fact[1] = 1
fact_inverse[0] = fact_inverse[1] = 1
for i in range(2,N+1):
    fact[i] = (fact[i-1]*i) % M
    fact_inverse[i] = pow(fact[i], M-2, M)
    
def binom(n, r):
    '''
    Calculate nCr = n! /(r! (n-r)!) % mod
    '''
    if n < r or n < 0 or r < 0:
        return 0
    else:
        return fact[n] * (fact_inverse[r] * fact_inverse[n-r] % M) % M

ans = 0

# Preprocess
# calculate 2**(2**n)
pow_pow2 = [0] * (N+1)
pow_pow2[0] = 2
for i in range(1, N+1):
    pow_pow2[i] = pow(pow_pow2[i-1], 2, M)

# (#ways to order i ramens, with each of n toppings chosen at most once)
ways2 = {n: {i: 0 for i in range(N+1)} for n in range(N+1)}
for n in range(N+1):
    ways = 0 # (#ways to choose n toppings, with each topping chosen at most once)
    temp = 0
    if n >= 1:
        for i in range(n+1):
            ways2[n][i] = (ways2[n][i] +  ways2[n-1][i]) % M # n-th topping is not used
            if i >= 1: ways2[n][i] = (ways2[n][i] + ways2[n-1][i-1]) % M # only n-th topping is used in an order
            ways2[n][i] = (ways2[n][i] + (i*ways2[n-1][i]) % M) % M # n-th topping is chosen with another topping
            temp = (temp + (ways2[n][i] * pow(2, (N-n)*i, M)) % M) % M # degree of freedom on each order is 2**(N-n)
    else: # n = 0
        ways2[n][0] = 1
        temp += ways2[n][0]
    ways = (temp * pow_pow2[N-n]) % M
    if n % 2 == 0:
        ans = (ans + (com.binom(N, n) * ways) % M) % M
    else:
        ans = (ans - (com.binom(N, n) * ways) % M) % M
print(ans)

Fixed submission:
from collections import defaultdict

class Combinatorics:
    def __init__(self, N, mod):
        '''
        Preprocess for calculating binomial coefficients nCr (0 <= r <= n, 0 <= n <= N)
        over the finite field Z/(mod)Z.
        Input:
            N (int): maximum n
            mod (int): a prime number. The order of the field Z/(mod)Z over which nCr is calculated.
        '''
        self.mod = mod
        self.fact = {i: None for i in range(N+1)}         # n!
        self.inverse = {i: None for i in range(1, N+1)}   # inverse of n in the field Z/(MOD)Z
        self.fact_inverse = {i: None for i in range(N+1)} # inverse of n! in the field Z/(MOD)Z
        
        # preprocess
        self.fact[0] = self.fact[1] = 1
        self.fact_inverse[0] = self.fact_inverse[1] = 1
        self.inverse[1] = 1
        for i in range(2, N+1):
            self.fact[i] = i * self.fact[i-1] % self.mod
            q, r = divmod(self.mod, i)
            self.inverse[i] = (- (q % self.mod) * self.inverse[r]) % self.mod
            self.fact_inverse[i] = self.inverse[i] * self.fact_inverse[i-1] % self.mod
    
    def binom(self, n, r):
        '''
        Calculate nCr = n! /(r! (n-r)!) % mod
        '''
        if n < r or n < 0 or r < 0:
            return 0
        else:
            return self.fact[n] * (self.fact_inverse[r] * self.fact_inverse[n-r] % self.mod) % self.mod

N, M = map(int, input().split())
com = Combinatorics(N, M)
ans = 0

# Preprocess
# calculate 2**n and 2**(2**n)
pow2 = [0] * (N*N // 4 + 1)
pow_pow2 = [0] * (N+1)
pow2[0] = 1; pow_pow2[0] = 2
for i in range(1, N*N // 4 + 1):
    pow2[i] = (pow2[i-1] * 2) % M
for i in range(1, N+1):
    pow_pow2[i] = pow(pow_pow2[i-1], 2, M)

# (#ways to order i ramens, with each of n toppings chosen at most once)
ways2 = {n: defaultdict(int) for n in range(N+1)}
for n in range(N+1):
    ways = 0 # (#ways to choose n toppings, with each topping chosen at most once)
    temp = 0
    if n >= 1:
        for i in range(n+1):
            ways2[n][i] = (ways2[n][i] +  ways2[n-1][i]) % M # n-th topping is not used
            if i >= 1: ways2[n][i] = (ways2[n][i] + ways2[n-1][i-1]) % M # only n-th topping is used in an order
            ways2[n][i] = (ways2[n][i] + (i*ways2[n-1][i]) % M) % M # n-th topping is chosen with another topping
            temp = (temp + (ways2[n][i] * pow2[(N-n)*i]) % M) % M # degree of freedom on each order is 2**(N-n)
    else: # n = 0
        ways2[n][0] = 1
        temp += ways2[n][0]
    ways = (temp * pow_pow2[N-n]) % M
    if n % 2 == 0:
        ans = (ans + (com.binom(N, n) * ways) % M) % M
    else:
        ans = (ans - (com.binom(N, n) * ways) % M) % M
print(ans)

Buggy python submission (verdict: Time Limit Exceeded):
class Combinatorics:
    def __init__(self, N, mod):
        '''
        Preprocess for calculating binomial coefficients nCr (0 <= r <= n, 0 <= n <= N)
        over the finite field Z/(mod)Z.
        Input:
            N (int): maximum n
            mod (int): a prime number. The order of the field Z/(mod)Z over which nCr is calculated.
        '''
        self.mod = mod
        self.fact = {i: None for i in range(N+1)}         # n!
        self.inverse = {i: None for i in range(1, N+1)}   # inverse of n in the field Z/(MOD)Z
        self.fact_inverse = {i: None for i in range(N+1)} # inverse of n! in the field Z/(MOD)Z
        
        # preprocess
        self.fact[0] = self.fact[1] = 1
        self.fact_inverse[0] = self.fact_inverse[1] = 1
        self.inverse[1] = 1
        for i in range(2, N+1):
            self.fact[i] = i * self.fact[i-1] % self.mod
            q, r = divmod(self.mod, i)
            self.inverse[i] = (- (q % self.mod) * self.inverse[r]) % self.mod
            self.fact_inverse[i] = self.inverse[i] * self.fact_inverse[i-1] % self.mod
    
    def perm(self, n, r):
        '''
        Calculate nPr = n! / (n-r)! % mod
        '''
        if n < r or n < 0 or r < 0:
            return 0
        else:
            return (self.fact[n] * self.fact_inverse[n-r]) % self.mod
    
    def binom(self, n, r):
        '''
        Calculate nCr = n! /(r! (n-r)!) % mod
        '''
        if n < r or n < 0 or r < 0:
            return 0
        else:
            return self.fact[n] * (self.fact_inverse[r] * self.fact_inverse[n-r] % self.mod) % self.mod
        
    def hom(self, n, r):
        '''
        Calculate nHr = {n+r-1}Cr % mod.
        Assign r objects to one of n classes.
        Arrangement of r circles and n-1 partitions:
            o o o | o o | | | o | | | o o | | o
        '''
        if n == 0 and r > 0:
            return 0
        if n >= 0 and r == 0:
            return 1
        return self.binom(n + r - 1, r)

N, M = map(int, input().split())
com = Combinatorics(N, M)

# Preprocess
# calculate 2**n and 2**(2**n)
pow2 = [0] * (N*N + 1)
pow_pow2 = [0] * (N+1)
pow2[0] = 1; pow_pow2[0] = 2
for i in range(1, N*N + 1):
    pow2[i] = (pow2[i-1] * 2) % M
for i in range(1, N+1):
    pow_pow2[i] = pow(pow_pow2[i-1], 2, M)

# (#ways to choose n toppings, with each topping chosen at most once)
ways = [0]*(N+1)
# (#ways to order i ramens, with each of n toppings chosen at most once)
ways2 = {n: {i: 0 for i in range(N+1)} for n in range(N+1)}
for n in range(N+1):
    temp = 0
    if n >= 1:
        for i in range(n+1):
            ways2[n][i] = (ways2[n][i] +  ways2[n-1][i]) % M # n-th topping is not used
            if i >= 1: ways2[n][i] = (ways2[n][i] + ways2[n-1][i-1]) % M # only n-th topping is used in an order
            ways2[n][i] = (ways2[n][i] + (i*ways2[n-1][i]) % M) % M # n-th topping is chosen with another topping
            temp = (temp + (ways2[n][i] * pow2[(N-n)*i]) % M) % M # degree of freedom on each order is 2**(N-n)
    else: # n = 0
        ways2[n][0] = 1
        temp += ways2[n][0]
    ways[n] = (temp * pow_pow2[N-n]) % M

ans = 0
for n in range(N+1):
    if n % 2 == 0:
        ans = (ans + (com.binom(N, n) * ways[n]) % M) % M
    else:
        ans = (ans - (com.binom(N, n) * ways[n]) % M) % M
print(ans)

Fixed submission:
from collections import defaultdict

class Combinatorics:
    def __init__(self, N, mod):
        '''
        Preprocess for calculating binomial coefficients nCr (0 <= r <= n, 0 <= n <= N)
        over the finite field Z/(mod)Z.
        Input:
            N (int): maximum n
            mod (int): a prime number. The order of the field Z/(mod)Z over which nCr is calculated.
        '''
        self.mod = mod
        self.fact = {i: None for i in range(N+1)}         # n!
        self.inverse = {i: None for i in range(1, N+1)}   # inverse of n in the field Z/(MOD)Z
        self.fact_inverse = {i: None for i in range(N+1)} # inverse of n! in the field Z/(MOD)Z
        
        # preprocess
        self.fact[0] = self.fact[1] = 1
        self.fact_inverse[0] = self.fact_inverse[1] = 1
        self.inverse[1] = 1
        for i in range(2, N+1):
            self.fact[i] = i * self.fact[i-1] % self.mod
            q, r = divmod(self.mod, i)
            self.inverse[i] = (- (q % self.mod) * self.inverse[r]) % self.mod
            self.fact_inverse[i] = self.inverse[i] * self.fact_inverse[i-1] % self.mod
    
    def binom(self, n, r):
        '''
        Calculate nCr = n! /(r! (n-r)!) % mod
        '''
        if n < r or n < 0 or r < 0:
            return 0
        else:
            return self.fact[n] * (self.fact_inverse[r] * self.fact_inverse[n-r] % self.mod) % self.mod

N, M = map(int, input().split())
com = Combinatorics(N, M)
ans = 0

# Preprocess
# calculate 2**n and 2**(2**n)
pow2 = [0] * (N*N // 4 + 1)
pow_pow2 = [0] * (N+1)
pow2[0] = 1; pow_pow2[0] = 2
for i in range(1, N*N // 4 + 1):
    pow2[i] = (pow2[i-1] * 2) % M
for i in range(1, N+1):
    pow_pow2[i] = pow(pow_pow2[i-1], 2, M)

# (#ways to order i ramens, with each of n toppings chosen at most once)
ways2 = {n: defaultdict(int) for n in range(N+1)}
for n in range(N+1):
    ways = 0 # (#ways to choose n toppings, with each topping chosen at most once)
    temp = 0
    if n >= 1:
        for i in range(n+1):
            ways2[n][i] = (ways2[n][i] +  ways2[n-1][i]) % M # n-th topping is not used
            if i >= 1: ways2[n][i] = (ways2[n][i] + ways2[n-1][i-1]) % M # only n-th topping is used in an order
            ways2[n][i] = (ways2[n][i] + (i*ways2[n-1][i]) % M) % M # n-th topping is chosen with another topping
            temp = (temp + (ways2[n][i] * pow2[(N-n)*i]) % M) % M # degree of freedom on each order is 2**(N-n)
    else: # n = 0
        ways2[n][0] = 1
        temp += ways2[n][0]
    ways = (temp * pow_pow2[N-n]) % M
    if n % 2 == 0:
        ans = (ans + (com.binom(N, n) * ways) % M) % M
    else:
        ans = (ans - (com.binom(N, n) * ways) % M) % M
print(ans)

Buggy python submission (verdict: Time Limit Exceeded):
class Combinatorics:
    def __init__(self, N, mod):
        '''
        Preprocess for calculating binomial coefficients nCr (0 <= r <= n, 0 <= n <= N)
        over the finite field Z/(mod)Z.
        Input:
            N (int): maximum n
            mod (int): a prime number. The order of the field Z/(mod)Z over which nCr is calculated.
        '''
        self.mod = mod
        self.fact = {i: None for i in range(N+1)}         # n!
        self.inverse = {i: None for i in range(1, N+1)}   # inverse of n in the field Z/(MOD)Z
        self.fact_inverse = {i: None for i in range(N+1)} # inverse of n! in the field Z/(MOD)Z
        
        # preprocess
        self.fact[0] = self.fact[1] = 1
        self.fact_inverse[0] = self.fact_inverse[1] = 1
        self.inverse[1] = 1
        for i in range(2, N+1):
            self.fact[i] = i * self.fact[i-1] % self.mod
            q, r = divmod(self.mod, i)
            self.inverse[i] = (- (q % self.mod) * self.inverse[r]) % self.mod
            self.fact_inverse[i] = self.inverse[i] * self.fact_inverse[i-1] % self.mod
    
    def perm(self, n, r):
        '''
        Calculate nPr = n! / (n-r)! % mod
        '''
        if n < r or n < 0 or r < 0:
            return 0
        else:
            return (self.fact[n] * self.fact_inverse[n-r]) % self.mod
    
    def binom(self, n, r):
        '''
        Calculate nCr = n! /(r! (n-r)!) % mod
        '''
        if n < r or n < 0 or r < 0:
            return 0
        else:
            return self.fact[n] * (self.fact_inverse[r] * self.fact_inverse[n-r] % self.mod) % self.mod
        
    def hom(self, n, r):
        '''
        Calculate nHr = {n+r-1}Cr % mod.
        Assign r objects to one of n classes.
        Arrangement of r circles and n-1 partitions:
            o o o | o o | | | o | | | o o | | o
        '''
        if n == 0 and r > 0:
            return 0
        if n >= 0 and r == 0:
            return 1
        return self.binom(n + r - 1, r)

N, M = map(int, input().split())
com = Combinatorics(N, M)

# Preprocess
ways = [0]*(N+1)
# (#ways to order i ramens, with each of n toppings chosen at most once)
ways2 = {n: {i: 0 for i in range(N+1)} for n in range(N+1)}
for n in range(N+1):
    temp = 0
    if n >= 1:
        for i in range(n+1):
            ways2[n][i] = (ways2[n][i] +  ways2[n-1][i]) % M # n-th topping is not used
            if i >= 1: ways2[n][i] = (ways2[n][i] + ways2[n-1][i-1]) % M # only n-th topping is used in an order
            ways2[n][i] = (ways2[n][i] + (i*ways2[n-1][i]) % M) % M # n-th topping is chosen with another topping
            temp = (temp + (ways2[n][i] * pow(2, (N-n) * i, M)) % M) % M # degree of freedom on each order is 2**(N-n)
    else: # n = 0
        ways2[n][0] = 1
        temp += ways2[n][0]
    ways[n] = temp * pow(2, 2**(N-n), M)

ans = 0
for n in range(N+1):
    if n % 2 == 0:
        ans = (ans + (com.binom(N, n) * ways[n]) % M) % M
    else:
        ans = (ans - (com.binom(N, n) * ways[n]) % M) % M
print(ans)

Fixed submission:
from collections import defaultdict

class Combinatorics:
    def __init__(self, N, mod):
        '''
        Preprocess for calculating binomial coefficients nCr (0 <= r <= n, 0 <= n <= N)
        over the finite field Z/(mod)Z.
        Input:
            N (int): maximum n
            mod (int): a prime number. The order of the field Z/(mod)Z over which nCr is calculated.
        '''
        self.mod = mod
        self.fact = {i: None for i in range(N+1)}         # n!
        self.inverse = {i: None for i in range(1, N+1)}   # inverse of n in the field Z/(MOD)Z
        self.fact_inverse = {i: None for i in range(N+1)} # inverse of n! in the field Z/(MOD)Z
        
        # preprocess
        self.fact[0] = self.fact[1] = 1
        self.fact_inverse[0] = self.fact_inverse[1] = 1
        self.inverse[1] = 1
        for i in range(2, N+1):
            self.fact[i] = i * self.fact[i-1] % self.mod
            q, r = divmod(self.mod, i)
            self.inverse[i] = (- (q % self.mod) * self.inverse[r]) % self.mod
            self.fact_inverse[i] = self.inverse[i] * self.fact_inverse[i-1] % self.mod
    
    def binom(self, n, r):
        '''
        Calculate nCr = n! /(r! (n-r)!) % mod
        '''
        if n < r or n < 0 or r < 0:
            return 0
        else:
            return self.fact[n] * (self.fact_inverse[r] * self.fact_inverse[n-r] % self.mod) % self.mod

N, M = map(int, input().split())
com = Combinatorics(N, M)
ans = 0

# Preprocess
# calculate 2**n and 2**(2**n)
pow2 = [0] * (N*N // 4 + 1)
pow_pow2 = [0] * (N+1)
pow2[0] = 1; pow_pow2[0] = 2
for i in range(1, N*N // 4 + 1):
    pow2[i] = (pow2[i-1] * 2) % M
for i in range(1, N+1):
    pow_pow2[i] = pow(pow_pow2[i-1], 2, M)

# (#ways to order i ramens, with each of n toppings chosen at most once)
ways2 = {n: defaultdict(int) for n in range(N+1)}
for n in range(N+1):
    ways = 0 # (#ways to choose n toppings, with each topping chosen at most once)
    temp = 0
    if n >= 1:
        for i in range(n+1):
            ways2[n][i] = (ways2[n][i] +  ways2[n-1][i]) % M # n-th topping is not used
            if i >= 1: ways2[n][i] = (ways2[n][i] + ways2[n-1][i-1]) % M # only n-th topping is used in an order
            ways2[n][i] = (ways2[n][i] + (i*ways2[n-1][i]) % M) % M # n-th topping is chosen with another topping
            temp = (temp + (ways2[n][i] * pow2[(N-n)*i]) % M) % M # degree of freedom on each order is 2**(N-n)
    else: # n = 0
        ways2[n][0] = 1
        temp += ways2[n][0]
    ways = (temp * pow_pow2[N-n]) % M
    if n % 2 == 0:
        ans = (ans + (com.binom(N, n) * ways) % M) % M
    else:
        ans = (ans - (com.binom(N, n) * ways) % M) % M
print(ans)