Problem:
Score : 600 points

Problem Statement

You are given N integers. The i-th integer is a_i.
Find the number, modulo 998244353, of ways to paint each of the integers red, green or blue so that the following condition is satisfied:

Let R, G and B be the sums of the integers painted red, green and blue, respectively. There exists a triangle with positive area whose sides have lengths R, G and B.

Constraints

3 \leq N \leq 300

1 \leq a_i \leq 300(1\leq i\leq N)

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
a_1
:
a_N

Output

Print the number, modulo 998244353, of ways to paint each of the integers red, green or blue so that the condition is satisfied.

Sample Input 1

4
1
1
1
2

Sample Output 1

18

We can only paint the integers so that the lengths of the sides of the triangle will be 1, 2 and 2, and there are 18 such ways.

Sample Input 2

6
1
3
2
3
5
2

Sample Output 2

150

Sample Input 3

20
3
1
4
1
5
9
2
6
5
3
5
8
9
7
9
3
2
3
8
4

Sample Output 3

563038556

Solution in Rust (1.15.1) (judge verdict: Wrong Answer)

#[allow(unused_imports)]
use std::cmp::*;
#[allow(unused_imports)]
use std::collections::*;

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}

fn main() {
    let n = read::<usize>();
    let mut a = vec![0usize; n];
    let big_int = 998244353i64;
    for i in 0..n {
        a[i] = read::<usize>();
    }

    let mut ans = 1i64;
    for i in 0..n {
        ans *= 3;
        ans %= big_int;
    }
    ans += 1;

    let mut temp = 1;
    for i in 0..n {
        temp *= 2;
        temp %= big_int;
    }
    temp *= 3;
    temp %= big_int;

    ans += big_int - temp;
    ans %= big_int;

    let fact = ModFactorial::new(n as i64, big_int);
    let fact = fact.fact;
    let inv_fact = (0..fact.len())
        .map(|x| mod_inverse(fact[x as usize], big_int))
        .collect::<Vec<_>>();
    let mut bins = Vec::with_capacity(n + 1);
    for i in 0..n + 1 {
        bins.push(ModBinominal::new(i as i64, big_int, &fact, &inv_fact).bin);
    }

    let mut sum_length = a.iter().sum::<usize>();
    let mut dp = vec![vec![0; n + 1]; sum_length + 1];

    for i in 0..n {
        let mut next_dp = dp.clone();
        for j in 0..sum_length + 1 {
            if j + a[i] > sum_length {
                continue;
            }
            for k in 1..n {
                next_dp[j + a[i]][k + 1] += dp[j][k];
            }
        }
        next_dp[a[i]][1] += 1;
        dp = next_dp;
    }
    let mut start_length;
    if sum_length % 2 == 0 {
        start_length = sum_length / 2;
    } else {
        start_length = sum_length / 2 + 1;
    }

    let mut temp2 = 0;
    for i in start_length..sum_length + 1 {
        for l1 in 0..n - 1 {
            let comb = dp[i][l1];
            if comb == 0 {
                continue;
            }
            for l2 in 1..n - 1 {
                if l1 + l2 > n - 1 {
                    continue;
                }
                let l3 = n - l1 - l2;
                temp2 += bins[l2 + l3][l2];
                temp2 %= big_int;
                temp2 *= comb;
                temp2 %= big_int;
            }
        }
    }
    temp2 *= 3;
    temp2 %= 3;
    ans += big_int - temp2;
    ans %= big_int;
    println!("{}", ans + 2);
    //println!("{}", temp2);
}

fn extgcd(a: i64, b: i64, x: &mut i64, y: &mut i64) -> i64 {
    let mut d = a;
    if b == 0 {
        *x = 1;
        *y = 0;
    } else {
        d = extgcd(b, a % b, y, x);
        *y -= (a / b) * (*x);
    }
    return d;
}

fn mod_inverse(a: i64, m: i64) -> i64 {
    let mut x: i64 = 0;
    let mut y: i64 = 0;
    extgcd(a, m, &mut x, &mut y);
    (m + x % m) % m
}

struct ModFactorial {
    fact: Vec<i64>,
}

// returns n! under mod m
impl ModFactorial {
    fn new(n: i64, m: i64) -> ModFactorial {
        let mut fact = vec![1; n as usize + 1];
        for i in 2..fact.len() {
            fact[i] = (i as i64 * fact[i - 1]) % m;
        }
        ModFactorial { fact: fact }
    }
}

// returns bin[i] = nCi under mod m
struct ModBinominal {
    bin: Vec<i64>,
}

impl ModBinominal {
    fn new(n: i64, m: i64, fact: &Vec<i64>, inv_fact: &Vec<i64>) -> ModBinominal {
        let mut bin = vec![1; n as usize + 1];
        if n == 0 {
            return ModBinominal { bin: bin };
        }
        for i in 1..n + 1 {
            let a2 = inv_fact[i as usize] % m;
            let a3 = inv_fact[(n - i) as usize] % m;
            bin[i as usize] = fact[n as usize] * ((a2 * a3) % m) % m;
        }
        ModBinominal { bin: bin }
    }
}