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;

#define mp make_pair
#define pb push_back

#define int int64_t
#define ld long double

const int MOD = 998244353;
const int N = 1e6+5;

mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());

int32_t main()
{
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);

    int n,s;
    cin>>n>>s;
    int dp[n+1][s+1];
    int arr[n];
    for(int i=0;i<n;i++) {
        cin>>arr[i];
    }

    dp[0][0]=1;
    memset(dp, 0, sizeof(dp));
    dp[0][0]=1;
    for(int i=0;i<n;i++) {
        for(int j=0;j<=s;j++) {
            dp[i+1][j]+=2*dp[i][j];
            dp[i+1][j]%=MOD;
            if(j+arr[i]<=s) {
                dp[i+1][j+arr[i]]+=dp[i][j];
                dp[i+1][j+arr[i]]%=MOD;
            }
        }
    }
    cout<<dp[n][s];

    return 0;
}