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 PyPy3 (7.3.0) (judge verdict: Time Limit Exceeded)

import sys
input=sys.stdin.readline

def calc():
  mod=998244353
  n,s=map(int,input().split())
  arr=list(map(int,input().split()))
  dp=[[[0]*(s+1) for _ in range(n+1)] for _ in range(n+1)]
  dp[0][0][0]=1
  for i in range(1,n+1):
    val=arr[i-1]
    for j in range(i,-1,-1):
      for k in range(s,-1,-1):
        dp[i][j][k]=dp[i-1][j][k]
        if k-val>=0 and dp[i-1][j-1][k-val]!=0:
          dp[i][j][k]+=dp[i-1][j-1][k-val]
  ans=0
  for i in range(1,n+1):
    ans+=dp[n][i][s]*pow(2,n-i,mod)
    ans%=mod
  print(ans)
calc()