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 Python (3.8.2) (judge verdict: Time Limit Exceeded)

import glob
import numpy as np
import copy

# 問題ごとのディレクトリのトップからの相対パス
REL_PATH = 'ABC\\169\\F'

# テスト用ファイル置き場のトップ
TOP_PATH = 'C:\\AtCoder'

class Common:

    problem = []
    index = 0

    def __init__(self, rel_path):
        self.rel_path = rel_path

    def initialize(self, path):
        file = open(path)
        self.problem = file.readlines()
        self.index = 0
        return

    def input_data(self):
        try:
            IS_TEST
            self.index += 1
            return self.problem[self.index-1]

        except NameError:
            return input()

    def resolve(self):
        pass

    def exec_resolve(self):
        try:
            IS_TEST
            for path in glob.glob(TOP_PATH + '\\' + self.rel_path + '/*.txt'):
                print("Test: " + path)
                self.initialize(path)
                self.resolve()
                print("\n\n")
        except NameError:
            self.resolve()


class Solver(Common):

    def resolve(self):

        tmp = self.input_data().split()
        N = int(tmp[0])
        S = int(tmp[1])
        tmp = self.input_data().split()
        A = [int(i) for i in tmp]
        A = np.array(A)
        A_sorted_index = np.argsort(A)

        # 集合全体に対してS=1-3000全ての組み合わせを格納するリスト
        combo_u_set = [[] for i in range(S+1)]
        combo_u_set[0].append(0)

        # 集合全体に対してS(1-3000)ができる組み合わせを網羅
        for i in A_sorted_index:
            a = A[i]
            for j in range(S, -1, -1):
                # 今確認している位置の数とaの合計が3000を超えればスルー
                if j + a > S:
                    continue
                # jの位置の組み合わせにiを足したものをj+aの位置に追加
                if j == 0:
                    pass
                l = copy.copy(combo_u_set[j])
                for k in range(len(l)):
                    l[k] += 1
                combo_u_set[j+a].extend(l)

        sum = 0
        # Sができる組み合わせが、何パターン作れるか合計
        for c in combo_u_set[S]:
            sum += 2 ** (N -c)
            sum = sum % 998244353

        print(str(sum))


solver = Solver(REL_PATH)
solver.exec_resolve()