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 C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;

int main() {
  int n, s;
  cin>>n>>s;
  int a[n];
  for (int i=0; i<n; i++) cin>>a[i];
  long long mod = 998244353;
  long long dp[s+1];
  for (int i=0; i<=s; i++) dp[i] = 0;
  dp[0] = 1;
  long long size = 1;
  for (int i=0; i<n; i++) {
    for (int j=s; j>0; j--) {
      if (j != a[i]) {
        dp[j] *= 2;
        dp[j] %= mod;
        if (j - a[i] >= 0) {
          dp[j] += dp[j-a[i]];
          dp[j] %= mod;
        }
      } else {
        dp[a[i]] *= 2;
        dp[a[i]] %= mod;
        dp[a[i]] += size;
        dp[a[i]] %= mod;
      }
    }
    // for (int i=0; i<=s; i++) {
    //   cout<<dp[i]<<" ";
    // }
    // cout<<endl;
    size *= 2;
    size %= mod;
  }
  cout<<dp[s]<<endl;
}