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 Rust (1.42.0) (judge verdict: Accepted)

#![allow(unused, non_snake_case, dead_code, non_upper_case_globals)]
use proconio::{fastout, input};
use std::cmp::*;
use std::collections::*;
use std::mem::*;

#[fastout]
fn main() {
  input! {
    n:usize,
    s:usize,
    a:[usize;n],
  }
  let mut dp = vec![vec![0; s + 1]; n + 1];
  dp[0][0] = 1;
  // for i in 0..=n {
  //   dp[i][0] = 1;
  // }
  for i in 0..n {
    // dp[i + 1][a[i]] += 1;
    // for j in 0..=s {
    //   dp[i + 1][j] += dp[i][j];
    // }
    for j in 0..=s {
      dp[i + 1][j] = (dp[i + 1][j] + 2 * dp[i][j]) % MOD;
      if j + a[i] <= s {
        dp[i + 1][j + a[i]] = (dp[i + 1][j + a[i]] + dp[i][j]) % MOD;
      }
    }
  }
  // dbg!(&dp);
  println!("{}", (dp[n][s] + MOD) % MOD);
}
const MOD: usize = 998244353;