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: Time Limit Exceeded)

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 ( *@) a b = (a * b) mod m in

    let module M = Map.Make (struct type t = int let compare = compare end) in
    let dp = Array.make (n + 1) M.empty in
    let () = dp.(0) <- M.add 0 1 M.empty in

    for i = 0 to n - 1 do
        let c = a.(i) in
        for j = i downto 0 do
            dp.(j + 1) <-
                M.fold (fun k v acc ->
                    if c + k > s then acc else
                    M.add (c + k) (v +@ try M.find (c + k) acc with _ -> 0) acc) dp.(j) dp.(j + 1)
        done
    done;
    let powmod a b =
        let rec loop r i acc =
            if i = 0 then acc else
                loop (r *@ r) (i lsr 1) (if i land 1 = 1 then acc *@ r else acc)
        in
        loop a b 1
    in
    let rec loop k acc =
        if k > n then acc else
            let acc =
                if M.mem s dp.(k) then acc +@ M.find s dp.(k) *@ powmod 2 (n - k) else acc
            in
            loop (k + 1) acc
    in
    loop 1 0 |> Printf.printf "%d\n"
)