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 Java (OpenJDK 11.0.6) (judge verdict: Time Limit Exceeded)

import java.util.ArrayList;
import java.util.HashSet;
import java.util.List;
import java.util.Scanner;
import java.util.Set;

public class Main {

	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		int N = sc.nextInt();
		int S = sc.nextInt();
		List<Set<Integer>> list = new ArrayList<Set<Integer>>();
		long A[][] = new long[N][S + 1];
		long sum = 0;
		long B[] = new long[3000];
		B[0] = 1;
		for (int i = 1; i < 3000; i++) {
			B[i] = (B[i - 1] * 2) % 998244353;
		}
		for (int i = 0; i < N; i++) {
			Set<Integer> se = new HashSet<Integer>();
			list.add(se);
			int temp = sc.nextInt();
			if (temp > S) {
				continue;
			}
			for (int j = i - 1; j >= 0; j--) {
				Set<Integer> set = list.get(j);
				Set<Integer> t1 = list.get(j + 1);
				for (Integer in : set) {
					if (in + temp <= S) {
						t1.add(in + temp);
						A[j + 1][in + temp] += A[j][in];
					}
				}
			}
			Set<Integer> zero = list.get(0);
			zero.add(temp);
			A[0][temp]++;
		}
		for (int i = 0; i < N; i++) {
			for (int j = 0; j < A[i][S]; j++) {
				sum = (sum + B[N - i - 1] % 998244353) % 998244353;
			}
		}
		System.out.println(sum);
	}
}