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_imports)]
#![allow(unused_macros)]
#![allow(non_snake_case)]
#![allow(dead_code)]
#![allow(unused_variables)]
#![allow(unused_mut)]
#![allow(unused_assignments)]
use proconio::input;
use proconio::marker::Usize1;
use std::collections::*;
use std::cmp::*;
use std::f64::consts::*;

const MOD: usize = 998244353;
const INF: usize = std::usize::MAX / 4;

fn main() {
    input! {
        N: usize,
        S: usize,
        A: [usize; N],
    }

    let mut dp = vec![0; S + 1];
    dp[0] = 1;

    for a in &A {
        let mut newDp: Vec<usize> = dp.iter().map(|d| d * 2).collect();
        for s in 0..(S + 1) {
            if s + a <= S {
                newDp[s + a] += dp[s];
            }
        }
        dp = newDp.into_iter().map(|d| d % MOD).collect();
    }

    println!("{}", dp[S]);
}