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 OCaml (4.10.0) (judge verdict: Accepted)

Scanf.scanf "%d %d" (fun n s ->
    let a = Array.init n (fun _ -> Scanf.scanf " %d" (fun a -> a)) in

    let m = 998244353 in
    let ( +@) a b = (a + b) mod m in

    let rec loop i dp =
        if i = n then dp.(s) else (
            let ndp = Array.map (fun a -> a +@ a) dp in
            let c = a.(i) in
            for j = 0 to s - c do
                ndp.(j + c) <- ndp.(j + c) +@ dp.(j)
            done;
            loop (i + 1) ndp
        )
    in
    let dp = Array.make (s + 1) 0 in
    let () = dp.(0) <- 1 in
    loop 0 dp |> Printf.printf "%d\n"
)