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 Ruby (2.7.1) (judge verdict: Time Limit Exceeded)

require 'set'; require 'prime'; require 'pp'
INF=Float::INFINITY; MOD=998244353

class Combi
  def initialize(max_n, mod = 10**9+7)
    @mod = mod
    @max_n = max_n

    @fac = Array.new(@max_n+1)
    @inv = Array.new(@max_n+1)
    @finv = Array.new(@max_n+1)

    @fac[0] = @fac[1] = 1
    @finv[0] = @finv[1] = 1
    @inv[1] = 1
    2.upto(@max_n) do |i|
      @fac[i] = @fac[i-1] * i % @mod
      @inv[i] = @mod - @inv[@mod%i] * (@mod/i) % @mod
      @finv[i] = @finv[i-1] * @inv[i] % @mod
    end
  end

  def nCk(n,k)
    @fac[n] * @finv[n - k] * @finv[k] % @mod
  end
end

n,s = gets.chomp.split.map(&:to_i)
a = gets.chomp.split.map(&:to_i)

h = Hash.new(0)
a.each do |e|
  h[e] += 1
end

def combination_sum2(cans, target)
  @cans = cans
  @target = target

  @cans.sort!
  @ans = Set.new(@cans.select{|can| can == target}.map{|can| [can]})
  @cans.select!{|can| can < target}
  @size = cans.size

  solve(0, [], 0)
  @ans.to_a
end

def solve(index, current, current_sum)
  index.upto(@size - 1) do |i|
    if current_sum + @cans[i] == @target
      @ans.add current + [@cans[i]]
    elsif current_sum + @cans[i] < @target
      solve(i + 1, current + [@cans[i]], current_sum + @cans[i])
    else
      break
    end
  end
end

combi = Combi.new(h.values.max+1, MOD)

ans = 0
comb = combination_sum2(a, s)
comb.each do |ar|
  ah = Hash.new(0)
  ar.each do |e|
    ah[e] += 1
  end
  anum = 1
  ah.each do |k, v|
    anum *= combi.nCk(h[k], v)
  end

  nokori = n - ar.size
  ans = (ans + anum * 2.pow(nokori, MOD)) % MOD
end

puts ans