Problem:
Score : 600 points

Problem Statement

Given are a sequence of N positive integers A_1, A_2, \ldots, A_N and another positive integer S.

For a non-empty subset T of the set \{1, 2, \ldots , N \}, let us define f(T) as follows:

f(T) is the number of different non-empty subsets \{x_1, x_2, \ldots , x_k \} of T such that A_{x_1}+A_{x_2}+\cdots +A_{x_k} = S.

Find the sum of f(T) over all 2^N-1 subsets T of \{1, 2, \ldots , N \}. Since the sum can be enormous, print it modulo 998244353.

Constraints

All values in input are integers.

1 \leq N \leq 3000

1 \leq S \leq 3000

1 \leq A_i \leq 3000

Input

Input is given from Standard Input in the following format:

N S
A_1 A_2 ... A_N

Output

Print the sum of f(T) modulo 998244353.

Sample Input 1

3 4
2 2 4

Sample Output 1

6

For each T, the value of f(T) is shown below. The sum of these values is 6.

f(\{1\}) = 0

f(\{2\}) = 0

f(\{3\}) = 1 (One subset \{3\} satisfies the condition.)

f(\{1, 2\}) = 1 (\{1, 2\})

f(\{2, 3\}) = 1 (\{3\})

f(\{1, 3\}) = 1 (\{3\})

f(\{1, 2, 3\}) = 2 (\{1, 2\}, \{3\})

Sample Input 2

5 8
9 9 9 9 9

Sample Output 2

0

Sample Input 3

10 10
3 1 4 1 5 9 2 6 5 3

Sample Output 3

3296

Solution in Python (3.8.2) (judge verdict: Runtime Error)

from collections import deque

def rec(dp, a, i, j, keiro, ans):
    if i == 0:
        if j == 0:
            ans.append(set(keiro))
        return
    
    if dp[i - 1][j] != 0:
        rec(dp, a, i - 1, j, keiro, ans)
    
    if j - a[i-1] >= 0 and dp[i - 1][j - a[i - 1]]:
        keiro.appendleft(i)
        rec(dp, a, i - 1, j - a[i-1], keiro, ans)
        keiro.popleft()

def main():
    n,s = map(int, input().split())
    a = list(map(int, input().split()))
    
    dp = [[0 for i in range(3010)] for j in range(n + 1)]
    dp[0][0] = 1
    for i in range(n):
        for j in range(3010):
            dp[i + 1][j] += dp[i][j]
            if (j + a[i] < 3010):
                dp[i + 1][j + a[i]] += dp[i][j]
    
    keiro = deque()
    ans = []
    
    rec(dp, a, n, s, keiro, ans)
    
    answer = 0
    
    for i in range(len(ans)):
        m = len(ans[i])
        add = 1
        for j in range(n - m):
            add *= 2
            add %= mod
        answer += add
        answer %= mod
    
    print(answer)

if __name__ == "__main__":
    mod = 998244353
    main()