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;
typedef long long ll;
const int MAXN = 3e3 + 5;
const int INF = 0x7f7f7f7f;
const ll p = 998244353;
#define DEBUG(x) cout << x << '\n'

int n, s, t, f[MAXN];
ll f_pow(ll a, ll b){
    ll res = 1;
    for(; b; b >>= 1, a = a * a % p)
        if(b & 1) res = res * a % p;
    return res;
}
int main(){
#ifdef Irene
    freopen("in.txt", "r", stdin);
#endif // Irene
    ios::sync_with_stdio(false);

    cin >> n >> s;
    f[0] = f_pow(2, n);
    ll inv2 = f_pow(2, p - 2);
    for(int i = 1; i <= n; i++){
        cin >> t;
        for(int j = s - t; j >= 0; j--)
            f[j+t] = (f[j+t] + f[j] * inv2) % p;
    }
    cout << f[s];
}