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 (1.15.1) (judge verdict: Accepted)

/*------------------------------------------------------------------*/
/*------------------------- begin template -------------------------*/
/*------------------------------------------------------------------*/

// ref: tanakh <https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8>
// diff: using Parser
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut parser = Parser::from_str($s);
        input_inner!{parser, $($r)*}
    };
    (parser = $parser:ident, $($r:tt)*) => {
        input_inner!{$parser, $($r)*}
    };
    (new_stdin_parser = $parser:ident, $($r:tt)*) => {
        let stdin = std::io::stdin();
        let reader = std::io::BufReader::new(stdin.lock());
        let mut $parser = Parser::new(reader);
        input_inner!{$parser, $($r)*}
    };
    ($($r:tt)*) => {
        input!{new_stdin_parser = parser, $($r)*}
    };
}

macro_rules! input_inner {
    ($parser:ident) => {};
    ($parser:ident, ) => {};
    ($parser:ident, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($parser, $t);
        input_inner!{$parser $($r)*}
    };
}

macro_rules! read_value {
    ($parser:ident, ( $($t:tt),* )) => {
        ( $(read_value!($parser, $t)),* )
    };
    ($parser:ident, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($parser, $t)).collect::<Vec<_>>()
    };
    ($parser:ident, chars) => {
        read_value!($parser, String).chars().collect::<Vec<char>>()
    };
    ($parser:ident, char_) => {
        read_value!($parser, String).chars().collect::<Vec<char>>()[0]
    };
    ($parser:ident, usize1) => {
        read_value!($parser, usize) - 1
    };
    ($parser:ident, i64_) => {
        $parser.fast_i64()
    };
    ($parser:ident, usize_) => {
        $parser.fast_i64() as usize
    };
    ($parser:ident, usize1_) => {
        ($parser.fast_i64() - 1) as usize
    };
    ($parser:ident, $t:ty) => {
        $parser.next::<$t>().expect("Parse error")
    };
}

use std::io;
use std::io::BufRead;
use std::str;

// ref: tatsuya6502 <https://qiita.com/tatsuya6502/items/cd448486f7ef7b5b8c7e>
// ref: wariuni <https://qiita.com/tatsuya6502/items/cd448486f7ef7b5b8c7e#comment-7040a5ae96305e884eb9>
// diff: using std::io::BufRead::fill_buf()
pub struct Parser<R> {
    pub reader: R,
    buf: Vec<u8>,
    pos: usize,
}

impl Parser<io::Empty> {
    pub fn from_str(s: &str) -> Parser<io::Empty> {
        Parser {
            reader: io::empty(),
            buf: s.as_bytes().to_vec(),
            pos: 0,
        }
    }
}

impl<R: BufRead> Parser<R> {
    pub fn new(reader: R) -> Parser<R> {
        Parser {
            reader: reader,
            buf: vec![],
            pos: 0,
        }
    }
    pub fn update_buf(&mut self) {
        self.buf.clear();
        self.pos = 0;
        loop {
            let (len, complete) = {
                let buf2 = self.reader.fill_buf().unwrap();
                self.buf.extend_from_slice(buf2);
                let len = buf2.len();
                (len, buf2[len - 1] <= 0x20)
            };
            self.reader.consume(len);
            if complete {
                break;
            }
        }
    }
    pub fn next<T: str::FromStr>(&mut self) -> Result<T, T::Err> {
        loop {
            let mut begin = self.pos;
            while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                begin += 1;
            }
            let mut end = begin;
            while end < self.buf.len() && (self.buf[end] > 0x20) {
                end += 1;
            }
            if begin != self.buf.len() {
                self.pos = end;
                return unsafe { str::from_utf8_unchecked(&self.buf[begin..end]) }.parse::<T>();
            } else {
                self.update_buf();
            }
        }
    }
    pub fn fast_i64(&mut self) -> i64 {
        loop {
            let mut begin = self.pos;
            while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                begin += 1;
            }
            if begin == self.buf.len() {
                self.update_buf();
                continue;
            }
            let mut res = 0;
            let (is_positive, mut end) = match self.buf[begin] {
                b'+' => (true, begin + 1),
                b'-' => (false, begin + 1),
                _ => (true, begin),
            };
            while end < self.buf.len() && (self.buf[end] > 0x20) {
                res = res * 10 + (self.buf[end] as i64 - '0' as i64);
                end += 1;
            }
            if begin != self.buf.len() {
                self.pos = end;
                return if is_positive { res } else { -res };
            } else {
                self.update_buf();
            }
        }
    }
}

#[allow(unused_imports)]
use std::cmp::*;
use std::collections::BTreeSet;
use std::marker::PhantomData;
use std::ops::*;

pub trait ModP {
    fn as_u64() -> u64;
}

#[allow(non_camel_case_types)]
#[derive(Clone, Copy, Debug)]
pub struct Mod_1_000_000_007();
impl ModP for Mod_1_000_000_007 {
    fn as_u64() -> u64 {
        1_000_000_007
    }
}

#[allow(non_camel_case_types)]
#[derive(Clone, Copy, Debug)]
pub struct Mod_998_244_353();
impl ModP for Mod_998_244_353 {
    fn as_u64() -> u64 {
        998_244_353
    }
}

#[derive(Clone, Copy, PartialEq, Debug)]
pub struct Modular<P: ModP + Copy>(u32, PhantomData<P>);

impl<P: ModP + Copy> Modular<P> {
    pub fn new(x: u32) -> Modular<P> {
        Modular((x as u64 % P::as_u64()) as u32, PhantomData)
    }
    pub fn pow(self, n: u64) -> Modular<P> {
        if n == 0 {
            return Modular(1, PhantomData);
        }
        let t = self.pow(n / 2);
        if n % 2 == 0 {
            t * t
        } else {
            t * t * self
        }
    }
    pub fn as_u32(self) -> u32 {
        self.0
    }
}

impl<P: ModP + Copy> Add for Modular<P> {
    type Output = Self;
    fn add(self, rhs: Self) -> Self::Output {
        let t = (self.0 + rhs.0) as u64;
        let u = if t < P::as_u64() { t } else { t - P::as_u64() };
        Modular(u as u32, PhantomData)
    }
}

impl<P: ModP + Copy> AddAssign for Modular<P> {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}

impl<P: ModP + Copy> Sub for Modular<P> {
    type Output = Self;
    fn sub(self, rhs: Self) -> Self::Output {
        let t = if self.0 >= rhs.0 {
            self.0 as u64 - rhs.0 as u64
        } else {
            self.0 as u64 + P::as_u64() - rhs.0 as u64
        };
        assert!(t < P::as_u64());
        Modular(t as u32, PhantomData)
    }
}

impl<P: ModP + Copy> SubAssign for Modular<P> {
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}

impl<P: ModP + Copy> Neg for Modular<P> {
    type Output = Modular<P>;
    fn neg(self) -> Modular<P> {
        Modular((P::as_u64() - self.0 as u64) as u32, PhantomData)
    }
}

impl<P: ModP + Copy> Mul for Modular<P> {
    type Output = Self;
    fn mul(self, rhs: Self) -> Self::Output {
        Modular(
            ((self.0 as u64 % P::as_u64()) * (rhs.0 as u64 % P::as_u64()) % P::as_u64()) as u32,
            PhantomData,
        )
    }
}

impl<P: ModP + Copy> MulAssign for Modular<P> {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}

impl<P: ModP + Copy> Div for Modular<P> {
    type Output = Self;
    fn div(self, rhs: Self) -> Self::Output {
        if rhs.0 == 0 {
            loop {}
        }
        assert!(rhs.0 != 0);
        self * rhs.pow(P::as_u64() - 2)
    }
}

pub fn gen_fact_table<P: ModP + Copy>(n: usize) -> Vec<Modular<P>> {
    let mut res = vec![Modular::<P>::new(1); n + 1];
    for i in 0..n {
        res[i + 1] = res[i] * Modular::<P>::new((i + 1) as u32);
    }
    res
}

pub fn gen_inv_table<P: ModP + Copy>(n: usize) -> Vec<Modular<P>> {
    let mut res = vec![Modular::<P>::new(0); n + 1];
    res[1] = Modular::<P>::new(1);
    for i in 2..n + 1 {
        res[i] =
            -res[P::as_u64() as usize % i] * Modular::<P>::new((P::as_u64() / i as u64) as u32);
    }
    res
}

fn main() {
    input! {
        new_stdin_parser = parser,
    }
    let res = solve(&mut parser);
    print!("{}", res);
}

/*------------------------------------------------------------------*/
/*-------------------------- end template --------------------------*/
/*------------------------------------------------------------------*/

type Mod = Modular<Mod_998_244_353>;

pub fn dfs(n: usize, m: usize, set: &mut BTreeSet<Vec<usize>>, buf: &mut Vec<usize>) {
    if m == 0 {
        set.insert(buf.clone());
        return;
    }
    for i in 0..n {
        for k in 0..n {
            if i == k {
                continue;
            }
            buf[i] += 1;
            buf[k] += 1;
            dfs(n, m - 1, set, buf);
            buf[i] -= 1;
            buf[k] -= 1;
        }
    }
}

pub fn f(n: usize, m: usize) -> usize {
    let mut buf = vec![0; n];
    let mut set = BTreeSet::new();
    dfs(n, m, &mut set, &mut buf);
    set.len()
}

use std::fmt::Write;
fn solve<R: BufRead>(parser: &mut Parser<R>) -> String {
    let mut buf = String::new();
    input! {
        parser = parser,
        n: usize,
        m: usize,
    }
    let facts = gen_fact_table::<Mod_998_244_353>(2000000);
    let mut res = Mod::new(0);
    let nCr = |n: usize, r: usize| {
        if n < r {
            Mod::new(0)
        } else {
            facts[n] / (facts[n - r] * facts[r])
        }
    };
    let nHr = |n: usize, r: usize| nCr(n + r - 1, n);
    for i in 0..m + 1 {
        if i % 2 == 1 {
            continue;
        }
        let s = nCr(n, m - i);
        let t = nHr(m + i / 2, n);
        let u = s * t;
        // eprintln!("i = {}, s = {}, t = {}, u = {}", i, s.as_u32(), t.as_u32(), u.as_u32());
        res += u;
    }
    for i in 0..m + 1 {
        if i % 2 == 1 {
            continue;
        }
        let s = nCr(n, m - i);
        let t = nHr(i / 2, n);
        let u = s * t;
        // eprintln!("i = {}, s = {}, t = {}, u = {}", i, s.as_u32(), t.as_u32(), u.as_u32());
        res -= Mod::new(n as u32) * u;
    }
    for i in 0..m + 1 {
        if i % 2 == 1 {
            continue;
        }
        let s = nCr(n - 1, m - i);
        let t = nHr(i / 2, n - 1);
        let u = s * t;
        // eprintln!("i = {}, s = {}, t = {}, u = {}", i, s.as_u32(), t.as_u32(), u.as_u32());
        res += Mod::new(n as u32) * u;
    }
    writeln!(buf, "{}", res.as_u32());
    buf
}