Problem:
Score : 900 points

Problem Statement

We have a sequence of N integers: x=(x_0,x_1,\cdots,x_{N-1}).
Initially, x_i=0 for each i (0 \leq i \leq N-1).

Snuke will perform the following operation exactly M times:

Choose two distinct indices i, j (0 \leq i,j \leq N-1,\ i \neq j).
Then, replace x_i with x_i+2 and x_j with x_j+1.

Find the number of different sequences that can result after M operations.
Since it can be enormous, compute the count modulo 998244353.

Constraints

2 \leq N \leq 10^6

1 \leq M \leq 5 \times 10^5

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N M

Output

Print the number of different sequences that can result after M operations, modulo 998244353.

Sample Input 1

2 2

Sample Output 1

3

After two operations, there are three possible outcomes:

x=(2,4)

x=(3,3)

x=(4,2)

For example, x=(3,3) can result after the following sequence of operations:

First, choose i=0,j=1, changing x from (0,0) to (2,1).

Second, choose i=1,j=0, changing x from (2,1) to (3,3).

Sample Input 2

3 2

Sample Output 2

19

Sample Input 3

10 10

Sample Output 3

211428932

Sample Input 4

100000 50000

Sample Output 4

3463133

Solution in Rust (1.42.0) (judge verdict: Accepted)

#![allow(unused_imports)]
#![allow(non_snake_case)]
use std::cmp::*;
use std::collections::*;
use std::io::Write;

#[allow(unused_macros)]
macro_rules! debug {
    ($($e:expr),*) => {
        #[cfg(debug_assertions)]
        $({
            let (e, mut err) = (stringify!($e), std::io::stderr());
            writeln!(err, "{} = {:?}", e, $e).unwrap()
        })*
    };
}

fn main() {
    let v = read_vec::<usize>();
    let (n, m) = (v[0], v[1]);
    let max_idx = 2 * m + n;
    let mut fact = vec![Modulo(1); max_idx + 1];
    for i in 1..max_idx + 1 {
        fact[i] = fact[i - 1] * Modulo(i as i64);
    }
    let fact_inv = fact.iter().map(|&x| x.inv()).collect::<Vec<_>>();

    let mut a = vec![Modulo(1); 2 * m + 1];
    for mi in 1..2 * m + 1 {
        a[mi] = mod_comb(mi + n - 2, mi, &fact, &fact_inv);
    }
    let mut b = vec![Modulo(0); 2 * m + 1];
    b[0] = a[0];
    for mi in 1..2 * m + 1 {
        b[mi] = b[mi - 1] + a[mi];
    }
    // debug!(a);

    let mut ans = Modulo(0);
    for p in 0..n + 1 {
        if p > m {
            break;
        }
        if p % 2 != m % 2 {
            continue;
        }
        let ncp = mod_comb(n, p, &fact, &fact_inv);
        let res = (3 * m - p) / 2;
        let all = mod_comb(res + n - 1, res, &fact, &fact_inv);
        let non1 = Modulo(p as i64) * (b[(m - p) / 2]);

        let non2;
        if (m - p) / 2 > 0 {
            non2 = Modulo((n - p) as i64) * (b[(m - p) / 2 - 1]);
        } else {
            non2 = Modulo(0);
        }
        ans += ncp * (all - non1 - non2);
    }
    println!("{}", ans);
}

#[derive(Clone, Copy, Debug, Default, Eq, Hash, Ord, PartialEq, PartialOrd)]
struct Modulo(i64);
static mut MODULUS: i64 = 998244353;
impl Modulo {
    fn set_modulus(m: i64) {
        unsafe {
            MODULUS = m;
        }
    }
    fn get_modulus() -> i64 {
        unsafe { MODULUS }
    }
    fn new(x: i64) -> Modulo {
        let m = Modulo::get_modulus();
        if x < 0 {
            Modulo(x % m + m)
        } else if x < m {
            Modulo(x)
        } else {
            Modulo(x % m)
        }
    }
    fn pow(self, p: i64) -> Modulo {
        if p == 0 {
            Modulo(1)
        } else {
            let mut t = self.pow(p / 2);
            t *= t;
            if p & 1 == 1 {
                t *= self;
            }
            t
        }
    }
    fn inv(self) -> Modulo {
        self.pow(Modulo::get_modulus() - 2)
    }
}
impl std::fmt::Display for Modulo {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        self.0.fmt(f)
    }
}
impl std::ops::AddAssign for Modulo {
    fn add_assign(&mut self, other: Modulo) {
        let m = Modulo::get_modulus();
        self.0 += other.0;
        if self.0 >= m {
            self.0 -= m;
        }
    }
}
impl std::ops::MulAssign for Modulo {
    fn mul_assign(&mut self, other: Modulo) {
        let m = Modulo::get_modulus();
        self.0 *= other.0;
        self.0 %= m;
    }
}
impl std::ops::SubAssign for Modulo {
    fn sub_assign(&mut self, other: Modulo) {
        let m = Modulo::get_modulus();
        self.0 += m - other.0;
        if self.0 >= m {
            self.0 -= m;
        }
    }
}
macro_rules! impl_modulo_ops {
    ($imp:ident, $method:ident, $assign_imp:ident, $assign_method:ident) => {
        impl<'a> std::ops::$assign_imp<&'a Modulo> for Modulo {
            fn $assign_method(&mut self, other: &'a Modulo) {
                std::ops::$assign_imp::$assign_method(self, *other);
            }
        }
        impl std::ops::$imp for Modulo {
            type Output = Modulo;
            fn $method(self, other: Modulo) -> Modulo {
                let mut x = self;
                std::ops::$assign_imp::$assign_method(&mut x, other);
                x
            }
        }
        impl<'a> std::ops::$imp<Modulo> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: Modulo) -> Modulo {
                std::ops::$imp::$method(*self, other)
            }
        }
        impl<'a> std::ops::$imp<&'a Modulo> for Modulo {
            type Output = Modulo;
            fn $method(self, other: &'a Modulo) -> Modulo {
                std::ops::$imp::$method(self, *other)
            }
        }
        impl<'a, 'b> std::ops::$imp<&'b Modulo> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: &'b Modulo) -> Modulo {
                std::ops::$imp::$method(*self, *other)
            }
        }
        impl std::ops::$assign_imp<i64> for Modulo {
            fn $assign_method(&mut self, other: i64) {
                std::ops::$assign_imp::$assign_method(self, Modulo::new(other));
            }
        }
        impl<'a> std::ops::$assign_imp<&'a i64> for Modulo {
            fn $assign_method(&mut self, other: &'a i64) {
                std::ops::$assign_imp::$assign_method(self, *other);
            }
        }
        impl std::ops::$imp<i64> for Modulo {
            type Output = Modulo;
            fn $method(self, other: i64) -> Modulo {
                let mut x = self;
                std::ops::$assign_imp::$assign_method(&mut x, other);
                x
            }
        }
        impl<'a> std::ops::$imp<&'a i64> for Modulo {
            type Output = Modulo;
            fn $method(self, other: &'a i64) -> Modulo {
                std::ops::$imp::$method(self, *other)
            }
        }
        impl<'a> std::ops::$imp<i64> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: i64) -> Modulo {
                std::ops::$imp::$method(*self, other)
            }
        }
        impl<'a, 'b> std::ops::$imp<&'b i64> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: &'b i64) -> Modulo {
                std::ops::$imp::$method(*self, *other)
            }
        }
    };
}
impl_modulo_ops!(Add, add, AddAssign, add_assign);
impl_modulo_ops!(Mul, mul, MulAssign, mul_assign);
impl_modulo_ops!(Sub, sub, SubAssign, sub_assign);

use std::iter::Sum;
impl Sum for Modulo {
    fn sum<I>(iter: I) -> Self
    where
        I: Iterator<Item = Modulo>,
    {
        iter.fold(Modulo(0), |a, b| a + b)
    }
}

impl<'a> Sum<&'a Modulo> for Modulo {
    fn sum<I>(iter: I) -> Self
    where
        I: Iterator<Item = &'a Self>,
    {
        iter.fold(Modulo(0), |a, b| a + b)
    }
}

fn mod_comb(n: usize, k: usize, fact: &[Modulo], fact_inv: &[Modulo]) -> Modulo {
    assert!(n >= k);
    fact[n] * fact[n - k].inv() * fact_inv[k]
}

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()
}