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F - Knapsack for All Subsets

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 $998244353$.

3 4
2 2 4

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.)
• $f(\{1, 2\}) = 1$ ($\{1, 2\}$)
• $f(\{2, 3\}) = 1$ ($\{3\}$)
• $f(\{1, 3\}) = 1$ ($\{3\}$)
• $f(\{1, 2, 3\}) = 2$ ($\{1, 2\}, \{3\}$)

5 8
9 9 9 9 9

0

10 10
3 1 4 1 5 9 2 6 5 3

3296