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: Runtime Error)

package com.company;
import java.util.*;
public class Main {
	static boolean[][] dp;
	static long ans = 0;
	static long len = 0;

	static long power(long x, long y, long p) {
		// Initialize result
		long res = 1;

		// Update x if it is more
		// than or equal to p
		x = x % p;

		if (x == 0) return 0; // In case x is divisible by p;

		while (y > 0) {
			// If y is odd, multiply x
			// with result
			if ((y & 1) == 1)
				res = (res * x) % p;

			// y must be even now
			// y = y / 2
			y = y >> 1;
			x = (x * x) % p;
		}
		return res;
	}

	static void display(ArrayList<Integer> v) {
		ans += power(2, len - v.size(), 998244353);
	}

	// A recursive function to print all subsets with the
	// help of dp[][]. Vector p[] stores current subset.
	static void printSubsetsRec(int arr[], int i, int sum, ArrayList<Integer> p) {
		// If we reached end and sum is non-zero. We print
		// p[] only if arr[0] is equal to sun OR dp[0][sum]
		// is true.
		if (i == 0 && sum != 0 && dp[0][sum]) {
			p.add(arr[i]);
			display(p);
			p.clear();
			return;
		}

		// If sum becomes 0
		if (i == 0 && sum == 0) {
			display(p);
			p.clear();
			return;
		}

		// If given sum can be achieved after ignoring
		// current element.
		if (dp[i - 1][sum]) {
			// Create a new vector to store path
			ArrayList<Integer> b = new ArrayList<>();
			b.addAll(p);
			printSubsetsRec(arr, i - 1, sum, b);
		}

		// If given sum can be achieved after considering
		// current element.
		if (sum >= arr[i] && dp[i - 1][sum - arr[i]]) {
			p.add(arr[i]);
			printSubsetsRec(arr, i - 1, sum - arr[i], p);
		}
	}

	// Prints all subsets of arr[0..n-1] with sum 0.
	static void printAllSubsets(int arr[], int n, int sum) {
		len = n;
		if (n == 0 || sum < 0)
			return;

		// Sum 0 can always be achieved with 0 elements
		dp = new boolean[n][sum + 1];
		for (int i = 0; i < n; ++i) {
			dp[i][0] = true;
		}

		// Sum arr[0] can be achieved with single element
		if (arr[0] <= sum)
			dp[0][arr[0]] = true;

		// Fill rest of the entries in dp[][]
		for (int i = 1; i < n; ++i)
			for (int j = 0; j < sum + 1; ++j)
				dp[i][j] = (arr[i] <= j) ? (dp[i - 1][j] ||
						dp[i - 1][j - arr[i]])
						: dp[i - 1][j];
		if (dp[n - 1][sum] == false) {
			return;
		}

		// Now recursively traverse dp[][] to find all
		// paths from dp[n-1][sum]
		ArrayList<Integer> p = new ArrayList<>();
		printSubsetsRec(arr, n - 1, sum, p);
	}

	public static void main(String args[]) {
		Scanner sc = new Scanner(System.in);
		int n = sc.nextInt();
		int k = sc.nextInt();
		int a[] = new int[n];
		for (int i = 0; i < n; i++) {
			a[i] = sc.nextInt();
		}
		printAllSubsets(a, n, k);
		System.out.println(ans % 998244353);
	}
}