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 C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading.Tasks;
using static System.Math;
public static class P
{
    public static void Main()
    {
        var ns = Console.ReadLine().Split().Select(int.Parse).ToArray();
        var n = ns[0];
        var s = ns[1];
        var a = Console.ReadLine().Split().Select(int.Parse).ToArray();
        //全ての部分集合について、
        //a個で和がbの個数
        ModInt[] dp = new ModInt[s + 1];
        dp[0] = Power(new ModInt(2), n);
        for (int i = 0; i < a.Length; i++)
        {
            var elem = a[i];
            for (int j = dp.Length - 1; j >= elem; j--)
            {
                dp[j] += dp[j - elem] / 2;
            }
        }
        Console.WriteLine(dp.Last());
    }

    static ModInt Power(ModInt n, long m)
    {
        ModInt pow = n;
        ModInt res = 1;
        while (m > 0)
        {
            if ((m & 1) == 1) res *= pow;
            pow *= pow;
            m >>= 1;
        }
        return res;
    }
}


struct ModInt
{
    public const int Mod = 998244353;
    const long POSITIVIZER = ((long)Mod) << 31;
    long Data;
    public ModInt(long data) { if ((Data = data % Mod) < 0) Data += Mod; }
    public static implicit operator long(ModInt modInt) => modInt.Data;
    public static implicit operator ModInt(long val) => new ModInt(val);
    public static ModInt operator +(ModInt a, int b) => new ModInt() { Data = (a.Data + b + POSITIVIZER) % Mod };
    public static ModInt operator +(ModInt a, long b) => new ModInt(a.Data + b);
    public static ModInt operator +(ModInt a, ModInt b) { long res = a.Data + b.Data; return new ModInt() { Data = res >= Mod ? res - Mod : res }; }
    public static ModInt operator -(ModInt a, int b) => new ModInt() { Data = (a.Data - b + POSITIVIZER) % Mod };
    public static ModInt operator -(ModInt a, long b) => new ModInt(a.Data - b);
    public static ModInt operator -(ModInt a, ModInt b) { long res = a.Data - b.Data; return new ModInt() { Data = res < 0 ? res + Mod : res }; }
    public static ModInt operator *(ModInt a, int b) => new ModInt(a.Data * b);
    public static ModInt operator *(ModInt a, long b) => a * new ModInt(b);
    public static ModInt operator *(ModInt a, ModInt b) => new ModInt() { Data = a.Data * b.Data % Mod };
    public static ModInt operator /(ModInt a, ModInt b) => new ModInt() { Data = a.Data * GetInverse(b) % Mod };
    public static bool operator ==(ModInt a, ModInt b) => a.Data == b.Data;
    public static bool operator !=(ModInt a, ModInt b) => a.Data != b.Data;
    public override string ToString() => Data.ToString();
    public override bool Equals(object obj) => (ModInt)obj == this;
    public override int GetHashCode() => (int)Data;
    static long GetInverse(long a)
    {
        long div, p = Mod, x1 = 1, y1 = 0, x2 = 0, y2 = 1;
        while (true)
        {
            if (p == 1) return x2 + Mod; div = a / p; x1 -= x2 * div; y1 -= y2 * div; a %= p;
            if (a == 1) return x1 + Mod; div = p / a; x2 -= x1 * div; y2 -= y1 * div; p %= a;
        }
    }
}