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>
#define watch(x) cout << (#x) << " is " << (x) << endl
const int mod = 998244353;

using namespace std;

int s, n;

void knapsackSubset()		//coeff x^s du polynome Pi(2+x^ai)
{	
	scanf("%d%d", &n, &s);
	int x;
	vector<long long> coeff(s+1, 0);
	cin >> x;
	coeff[0] = 2;
	if (x <= s)
	 coeff[x] = 1;
	for(int i = 1; i < n; i++)
	{
		scanf("%d", &x);
		vector<long long>oldCo = coeff; //ancient coordonnes, (ne pas oublier qu'on mulitplie par 2, on doit se rappeler des ancierts coo, quitte a diviser par 2 aussi)
		for (int j = 0; j <= s; j++)
		{
			coeff[j] = (2*coeff[j])%mod;
			if (j >= x)
				coeff[j] += oldCo[j-x];
			coeff[j] %= mod;
		}
	} 
	cout << coeff[s] << endl;	
}

int main()
{
	#ifdef ME
		freopen("input.txt", "r", stdin);
		freopen("output.txt", "w", stdout);
	#endif

	knapsackSubset();
	return 0;
}