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 C++14 (GCC 5.4.1) (judge verdict: Accepted)

//08:36~
#include <bits/stdc++.h>
using namespace std;

template <uint32_t mod>
class modint{
    uint64_t value;
public:
    constexpr modint(const int64_t x=0) noexcept: value(x % mod + (x < 0 ? mod : 0)){ }
    constexpr explicit operator uint64_t() const noexcept{ return value; }
    constexpr modint inverse() const noexcept{ return pow(*this, mod-2); }
    constexpr bool operator==(const modint &rhs) const noexcept{ return value == rhs.value; }
    constexpr bool operator!=(const modint &rhs) const noexcept{ return value != rhs.value; }
    constexpr modint operator+() const noexcept{ return modint(*this); }
    constexpr modint operator-() const noexcept{ return modint(mod - value); }
    constexpr modint operator+(const modint &rhs) const noexcept{ return modint(*this) += rhs; }
    constexpr modint operator-(const modint &rhs) const noexcept{ return modint(*this) -= rhs; }
    constexpr modint operator*(const modint &rhs) const noexcept{ return modint(*this) *= rhs; }
    constexpr modint operator/(const modint &rhs) const noexcept{ return modint(*this) /= rhs; }
    constexpr modint &operator+=(const modint &rhs) noexcept{
        if((value += rhs.value) >= mod) value -= mod;
        return *this;
    }
    constexpr modint &operator-=(const modint &rhs) noexcept{ return *this += mod - rhs.value; }
    constexpr modint &operator*=(const modint &rhs) noexcept{
        if((value *= rhs.value) >= mod) value %= mod;
        return *this;
    }
    constexpr modint &operator/=(const modint &rhs) noexcept{ return *this *= rhs.inverse(); }
    constexpr modint operator++(int) noexcept{
        modint ret(*this);
        if((++value) >= mod) value -= mod;
        return ret;
    }
    constexpr modint operator--(int) noexcept{
        modint ret(*this);
        if((value += mod - 1) >= mod) value -= mod;
        return ret;
    }
    constexpr modint &operator++() noexcept{ return *this += 1; }
    constexpr modint &operator--() noexcept{ return *this -= 1; }
    friend std::ostream &operator<<(std::ostream &os, const modint<mod> &x){ return os << x.value; }
    friend std::istream &operator>>(std::istream &is, modint<mod> &x){
        int64_t i;
        is >> i;
        x = modint<mod>(i);
        return is;
    }
    friend constexpr modint<mod> pow(const modint<mod> &x, uint64_t y){
        modint<mod> ret{1}, m{x};
        while(y > 0){
            if(y & 1) ret *= m;
            m *= m;
            y >>= 1;
        }
        return ret;
    }
};
constexpr int64_t mod = 998244353;

int main(){
    cin.tie(nullptr);
    ios::sync_with_stdio(false);
    constexpr int fmax = 3000000;
    static modint<mod> fact[fmax+1], finv[fmax+1];
    fact[0] = 1;
    for(int i=0;i<fmax;++i) fact[i+1] = fact[i] * (i+1);
    finv[fmax] = fact[fmax].inverse();
    for(int i=fmax;i>0;--i) finv[i-1] = finv[i] * i;
    auto comb = [&](int n, int r){ return fact[n] * finv[r] * finv[n-r]; };

    int n, m;
    cin >> n >> m;
    modint<mod> ans = comb(3*m + n-1, n-1);
    for(int o=m+1;o<=n;++o){
        if(3*m < o) break;
        if((3*m-o) % 2 != 0) continue;
        ans -= comb(n, o) * comb((3*m-o)/2 + n-1, n-1);
    }
    for(int x=2*m+1;x<=3*m;++x){
        ans -= modint<mod>{n} * comb(3*m-x + n-2, n-2);
    }
    cout << ans << endl;
    return 0;
}