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: Runtime Error)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using static System.Math;

namespace CSharpSpiral {
	class Program {
		static void Main(string[] args) {
			var sw = new StreamWriter(Console.OpenStandardOutput()) {
				AutoFlush=
#if DEBUG
			true
#else
			false
#endif
			};
			Console.SetOut(sw);
			new Program().Solve(new ConsoleInput(
#if DEBUG
			new StreamReader("../../../inputData.txt")
#else
			Console.In
#endif
			, ' '));
			Console.Out.Flush();
		}
		public void Solve(ConsoleInput cin) {
			var F = new QuotientRing(998244353);
			const int modulo = 998244353;
			var N = cin.ReadInt;
			var S = cin.ReadInt;
			var A = cin.ReadIntArray(N);
			var dp = new long[N+1][];
			for(int i = 0; i<N+1; i++) {
				dp[i]=new long[S+1];
			}
			dp[0][0]=1;
			for(int i = 0; i<N; i++) {
				var a = A[i];
				for(int s = 0; s<a; s++) {
					dp[i+1][s]=dp[i][s]*2;
					if(dp[i+1][s]>=modulo) dp[i+1][s]%=modulo;
				}
				for(int s = 0; s+a<=S; s++) {
					long below = dp[i][s]+dp[i][s+a]*2;
					if(below>=modulo) below%=modulo;
					dp[i+1][s+a]=below;
				}
			}
			Console.WriteLine(dp[N][S]);
		}

	}
	static class ExtendClass {
		public static string Join<T>(this IEnumerable<T> X, string separator = " ") => string.Join(separator, X);
	}
	class ConsoleInput {
		private readonly TextReader _stream;
		private readonly char _separator = ' ';
		private readonly Queue<string> inputStream;
		public ConsoleInput(TextReader stream, char separator = ' ') {
			this._separator=separator;
			this._stream=stream;
			inputStream=new Queue<string>();
		}
		public string Read {
			get {
				if(inputStream.Count!=0) return inputStream.Dequeue();
				string[] tmp = _stream.ReadLine().Split(_separator);
				for(int i = 0; i<tmp.Length; ++i)
					inputStream.Enqueue(tmp[i]);
				return inputStream.Dequeue();
			}
		}
		public string ReadLine { get { return _stream.ReadLine(); } }
		public int ReadInt { get { return int.Parse(Read); } }
		public long ReadLong { get { return long.Parse(Read); } }
		public double ReadDouble { get { return double.Parse(Read); } }
		public string[] ReadStrArray(long N) {
			var ret = new string[N];
			for(long i = 0; i<N; ++i)
				ret[i]=Read;
			return ret;
		}
		public int[] ReadIntArray(long N) { var ret = new int[N]; for(long i = 0; i<N; ++i) ret[i]=ReadInt; return ret; }
		public long[] ReadLongArray(long N) { var ret = new long[N]; for(long i = 0; i<N; ++i) ret[i]=ReadLong; return ret; }
		public double[] ReadDoubleArray(long N) { var ret = new double[N]; for(long i = 0; i<N; ++i) ret[i]=ReadDouble; return ret; }
		public bool isEOF {
			get { return _stream.Peek()==-1; }
		}
	}
	class QuotientRing {
		public readonly long modulo;
		public QuotientRing(int mModulo = 1000*1000*1000+7) {
			modulo=mModulo;
		}
		private long[] factorialTable = { 1 };
		private int factorialTableLastIndex { get { return factorialTable.Length-1; } }
		public long mod(long x) => x<0 ? x%modulo+modulo : x%modulo;
		public long Add(long a, long b) => mod(a+b);
		public long Sub(long a, long b) => mod(a-b);
		public long Mul(long a, long b) => mod(a*b);
		public long Pow(long a, long n) {
			if(n==0) return 1;
			var R = 1L;
			n=n<0 ? n%(modulo-1)+(modulo-1) : n%(modulo-1);
			for(; n>0; a=Mul(a, a), n>>=1) {
				if((n&1)!=0) R=Mul(R, a);
			}
			return R;
		}
		private void ExpandFactorialTable(int N) {
			var k = factorialTableLastIndex;
			Array.Resize(ref factorialTable, N+1);
			for(int i = k+1; i<=N; i++) {
				if(i==0) { factorialTable[i]=1; continue; }
				factorialTable[i]=Mul(factorialTable[i-1], i);
			}
		}
		public long Factorial(int n) {
			if(n>factorialTableLastIndex)
				ExpandFactorialTable(n);
			return factorialTable[n];
		}
		public long Permutation(int n, int k) {
			if(k<0||k>n) return 0;
			var Res = 1L;
			for(int i = 0; i<k; i++)
				Res=Mul(Res, n-i);
			return Res;
		}
		public virtual long Calculate(params object[] RPN) {
			var S = new Stack<long>();
			foreach(var token in RPN) {
				var type = token.GetType();
				if(type==typeof(int)||type==typeof(long)) {
					S.Push((long)token);
				} else if(type==typeof(char)) {
					long v1 = 0L, v2 = 0L;
					v2=S.Pop();
					HashSet<char> unaries = new HashSet<char>() { '!' };
					if(!unaries.Contains((char)token))
						v1=S.Pop();
					long res;
					switch((char)token) {
						case '+': res=Add(v1, v2); break;
						case '-': res=Sub(v1, v2); break;
						case '*': res=Mul(v1, v2); break;
						case '^': res=Pow(v1, v2); break;
						case '!': res=Factorial((int)v2); break;
						default: throw new InvalidOperationException();
					}
					S.Push(res);
				} else throw new InvalidOperationException();
			}
			return S.Peek();
		}
	}
}