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 Go (1.14.1) (judge verdict: Time Limit Exceeded)

package main

import "fmt"

const mod = 998244353

func power(a, b int) int {
	if b == 0 {
		return 1
	}
	ret := power(a, b/2)
	ret *= ret
	if b%2 == 1 {
		ret *= a
	}
	return ret
}

func solve(n, s, j int, a []int) int {
	if n == 1 {
		if s == a[0] {
			return power(2, j)
		} else if s == 0 {
			return power(2, j+1)
		}
		return 0
	}
	return (solve(n-1, s-a[n-1], j, a) + solve(n-1, s, j+1, a)) % mod
}

func main() {
	var n, s int
	fmt.Scan(&n, &s)
	a := make([]int, n)
	for i := 0; i < n; i++ {
		fmt.Scan(&a[i])
	}
	fmt.Println(solve(n, s, 0, a))
}