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: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Linq;
using CompLib.Collections;
using CompLib.Mathematics;
using CompLib.Util;

public class Program
{
    int N;
    int S;
    int[] A;
    public void Solve()
    {
        var sc = new Scanner();
        N = sc.NextInt();
        S = sc.NextInt();
        A = sc.IntArray();

        // 大きさ、合計
        HashMap<int, ModInt>[] map = new HashMap<int, ModInt>[N + 1];
        for (int i = 0; i <= N; i++)
        {
            map[i] = new HashMap<int, ModInt>();
        }
        map[0][0] = 1;

        // N
        for (int i = 0; i < N; i++)
        {
            for (int j = i; j >= 0; j--)
            {
                foreach (var p in map[j])
                {
                    if (p.Key + A[i] <= S) map[j + 1][p.Key + A[i]] += p.Value;
                }
            }
        }
        ModInt ans = 0;
        for (int i = 0; i <= N; i++)
        {
            ans += ModInt.Pow(2, N - i) * map[i][S];
        }
        Console.WriteLine(ans);
    }

    public static void Main(string[] args) => new Program().Solve();
}


namespace CompLib.Collections
{
    using System.Collections.Generic;
    public class HashMap<TKey, TValue> : Dictionary<TKey, TValue>
    {
        public new TValue this[TKey key]
        {
            get
            {
                TValue o;
                return TryGetValue(key, out o) ? o : default(TValue);
            }
            set { base[key] = value; }
        }
    }
}

namespace CompLib.Mathematics
{
    #region ModInt
    /// <summary>
    /// [0,<see cref="Mod"/>) までの値を取るような数
    /// </summary>
    public struct ModInt
    {
        /// <summary>
        /// 剰余を取る値．
        /// </summary>
        public const long Mod = 998244353;

        /// <summary>
        /// 実際の数値．
        /// </summary>
        public long num;
        /// <summary>
        /// 値が <paramref name="n"/> であるようなインスタンスを構築します．
        /// </summary>
        /// <param name="n">インスタンスが持つ値</param>
        /// <remarks>パフォーマンスの問題上，コンストラクタ内では剰余を取りません．そのため，<paramref name="n"/> ∈ [0,<see cref="Mod"/>) を満たすような <paramref name="n"/> を渡してください．このコンストラクタは O(1) で実行されます．</remarks>
        public ModInt(long n) { num = n; }
        /// <summary>
        /// このインスタンスの数値を文字列に変換します．
        /// </summary>
        /// <returns>[0,<see cref="Mod"/>) の範囲内の整数を 10 進表記したもの．</returns>
        public override string ToString() { return num.ToString(); }
        public static ModInt operator +(ModInt l, ModInt r) { l.num += r.num; if (l.num >= Mod) l.num -= Mod; return l; }
        public static ModInt operator -(ModInt l, ModInt r) { l.num -= r.num; if (l.num < 0) l.num += Mod; return l; }
        public static ModInt operator *(ModInt l, ModInt r) { return new ModInt(l.num * r.num % Mod); }
        public static implicit operator ModInt(long n) { n %= Mod; if (n < 0) n += Mod; return new ModInt(n); }

        /// <summary>
        /// 与えられた 2 つの数値からべき剰余を計算します．
        /// </summary>
        /// <param name="v">べき乗の底</param>
        /// <param name="k">べき指数</param>
        /// <returns>繰り返し二乗法により O(N log N) で実行されます．</returns>
        public static ModInt Pow(ModInt v, long k) { return Pow(v.num, k); }

        /// <summary>
        /// 与えられた 2 つの数値からべき剰余を計算します．
        /// </summary>
        /// <param name="v">べき乗の底</param>
        /// <param name="k">べき指数</param>
        /// <returns>繰り返し二乗法により O(N log N) で実行されます．</returns>
        public static ModInt Pow(long v, long k)
        {
            long ret = 1;
            for (k %= Mod - 1; k > 0; k >>= 1, v = v * v % Mod)
                if ((k & 1) == 1) ret = ret * v % Mod;
            return new ModInt(ret);
        }
        /// <summary>
        /// 与えられた数の逆元を計算します．
        /// </summary>
        /// <param name="v">逆元を取る対象となる数</param>
        /// <returns>逆元となるような値</returns>
        /// <remarks>法が素数であることを仮定して，フェルマーの小定理に従って逆元を O(log N) で計算します．</remarks>
        public static ModInt Inverse(ModInt v) { return Pow(v, Mod - 2); }
    }
    #endregion
    #region Binomial Coefficient
    public class BinomialCoefficient
    {
        public ModInt[] fact, ifact;
        public BinomialCoefficient(int n)
        {
            fact = new ModInt[n + 1];
            ifact = new ModInt[n + 1];
            fact[0] = 1;
            for (int i = 1; i <= n; i++)
                fact[i] = fact[i - 1] * i;
            ifact[n] = ModInt.Inverse(fact[n]);
            for (int i = n - 1; i >= 0; i--)
                ifact[i] = ifact[i + 1] * (i + 1);
            ifact[0] = ifact[1];
        }
        public ModInt this[int n, int r]
        {
            get
            {
                if (n < 0 || n >= fact.Length || r < 0 || r > n) return 0;
                return fact[n] * ifact[n - r] * ifact[r];
            }
        }
        public ModInt RepeatedCombination(int n, int k)
        {
            if (k == 0) return 1;
            return this[n + k - 1, k];
        }
    }
    #endregion
}


namespace CompLib.Util
{
    using System;
    using System.Linq;

    class Scanner
    {
        private string[] _line;
        private int _index;
        private const char Separator = ' ';

        public Scanner()
        {
            _line = new string[0];
            _index = 0;
        }

        public string Next()
        {
            while (_index >= _line.Length)
            {
                _line = Console.ReadLine().Split(Separator);
                _index = 0;
            }

            return _line[_index++];
        }

        public int NextInt() => int.Parse(Next());
        public long NextLong() => long.Parse(Next());
        public double NextDouble() => double.Parse(Next());
        public decimal NextDecimal() => decimal.Parse(Next());
        public char NextChar() => Next()[0];
        public char[] NextCharArray() => Next().ToCharArray();

        public string[] Array()
        {
            _line = Console.ReadLine().Split(Separator);
            _index = _line.Length;
            return _line;
        }

        public int[] IntArray() => Array().Select(int.Parse).ToArray();
        public long[] LongArray() => Array().Select(long.Parse).ToArray();
        public double[] DoubleArray() => Array().Select(double.Parse).ToArray();
        public decimal[] DecimalArray() => Array().Select(decimal.Parse).ToArray();
    }
}