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 Python (3.8.2) (judge verdict: Accepted)

import sys
input = sys.stdin.readline
import numpy as np
from numba import njit


def read():
    N, S = map(int, input().strip().split())
    A = np.fromstring(input().strip(), dtype=np.int32, sep=" ")
    return N, S, A


@njit
def solve(N, S, A, MOD=998244353):
    dp = np.zeros((S+1), dtype=np.int32)
    dp[0] = 1
    for i in range(N):
        for j in range(S, -1, -1):
            v = 2 * dp[j]
            if j - A[i] >= 0:
                v += dp[j-A[i]]
            dp[j] = v % MOD
    return dp[S]


if __name__ == '__main__':
    inputs = read()
    outputs = solve(*inputs)
    if outputs is not None:
        print("%s" % str(outputs))