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: Wrong Answer)

using System;
using System.Collections;
using System.Collections.Generic;
using System.IO;
using System.Linq;

namespace Tasks
{
    public class F
    {
        static void Main(string[] args)
        {
            var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
            Console.SetOut(sw);
            Solve();
            Console.Out.Flush();
        }

        public static void Solve()
        {
            var (N, S) = Scanner.Scan<int, int>();
            var A = Scanner.ScanEnumerable<int>().Select(x => (ModuloInteger)x).ToArray();
            var dp = new ModuloInteger[N + 1][].Select(x => new ModuloInteger[S + 1]).ToArray();
            dp[0][0] = 1;
            for (var i = 0; i < N; i++)
            {
                for (var j = 0; j <= S; j++)
                {
                    dp[i + 1][j] += dp[i][j] * 2;
                    if (j + A[i] <= S) dp[i + 1][j + A[i]] += dp[i][j];
                }
            }

            Console.WriteLine(string.Join("\n", dp.Select(x => string.Join(" ", x))));

            Console.WriteLine(dp[N][S]);
        }

        public static class Scanner
        {
            private static Queue<string> queue = new Queue<string>();
            public static T Next<T>()
            {
                if (!queue.Any()) foreach (var item in Console.ReadLine().Trim().Split(" ")) queue.Enqueue(item);
                return (T)Convert.ChangeType(queue.Dequeue(), typeof(T));
            }
            public static T Scan<T>() => Next<T>();
            public static (T1, T2) Scan<T1, T2>() => (Next<T1>(), Next<T2>());
            public static (T1, T2, T3) Scan<T1, T2, T3>() => (Next<T1>(), Next<T2>(), Next<T3>());
            public static (T1, T2, T3, T4) Scan<T1, T2, T3, T4>() => (Next<T1>(), Next<T2>(), Next<T3>(), Next<T4>());
            public static (T1, T2, T3, T4, T5) Scan<T1, T2, T3, T4, T5>() => (Next<T1>(), Next<T2>(), Next<T3>(), Next<T4>(), Next<T5>());
            public static IEnumerable<T> ScanEnumerable<T>() => Console.ReadLine().Trim().Split(" ").Select(x => (T)Convert.ChangeType(x, typeof(T)));
        }

        public readonly struct ModuloInteger
        {
            public long Value { get; }
            public const long Modulo = 998244353;
            public ModuloInteger(long data) => Value = (0 <= data ? data : data + Modulo) % Modulo;
            public static implicit operator long(ModuloInteger mint) => mint.Value;
            public static implicit operator int(ModuloInteger mint) => (int)mint.Value;
            public static implicit operator ModuloInteger(long val) => new ModuloInteger(val);
            public static implicit operator ModuloInteger(int val) => new ModuloInteger(val);
            public static ModuloInteger operator +(ModuloInteger a, ModuloInteger b) => a.Value + b.Value;
            public static ModuloInteger operator +(ModuloInteger a, long b) => a.Value + b;
            public static ModuloInteger operator +(ModuloInteger a, int b) => a.Value + b;
            public static ModuloInteger operator -(ModuloInteger a, ModuloInteger b) => a.Value - b.Value;
            public static ModuloInteger operator -(ModuloInteger a, long b) => a.Value - b;
            public static ModuloInteger operator -(ModuloInteger a, int b) => a.Value - b;
            public static ModuloInteger operator *(ModuloInteger a, ModuloInteger b) => a.Value * b.Value;
            public static ModuloInteger operator *(ModuloInteger a, long b) => a.Value * (b % Modulo);
            public static ModuloInteger operator *(ModuloInteger a, int b) => a.Value * (b % Modulo);
            public static ModuloInteger operator /(ModuloInteger a, ModuloInteger b) => a * b.Inverse();
            public static ModuloInteger operator /(ModuloInteger a, long b) => a.Value * Inverse(b);
            public static ModuloInteger operator /(ModuloInteger a, int b) => a.Value * Inverse(b);
            public static bool operator ==(ModuloInteger a, ModuloInteger b) => a.Value == b.Value;
            public static bool operator !=(ModuloInteger a, ModuloInteger b) => a.Value != b.Value;
            public bool Equals(ModuloInteger other) => Value == other.Value;
            public override bool Equals(object obj) => obj is ModuloInteger other && Equals(other);
            public override int GetHashCode() => Value.GetHashCode();
            public override string ToString() => Value.ToString();
            public ModuloInteger Inverse() => Inverse(Value);
            public static ModuloInteger Inverse(long a)
            {
                if (a == 0) return 0;
                var p = Modulo;
                var (x1, y1, x2, y2) = (1L, 0L, 0L, 1L);
                while (true)
                {
                    if (p == 1) return (x2 % Modulo + Modulo) % Modulo;
                    var div = a / p;
                    x1 -= x2 * div;
                    y1 -= y2 * div;
                    a %= p;
                    if (a == 1) return (x1 % Modulo + Modulo) % Modulo;
                    div = p / a;
                    x2 -= x1 * div;
                    y2 -= y1 * div;
                    p %= a;
                }
            }
            public ModuloInteger Power(long n) => Power(Value, n);
            public static ModuloInteger Power(ModuloInteger x, long n)
            {
                if (n < 0) throw new ArgumentException();
                var r = new ModuloInteger(1);
                while (n > 0)
                {
                    if ((n & 1) > 0) r *= x;
                    x *= x;
                    n >>= 1;
                }
                return r;
            }
        }
    }
}