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 PyPy3 (2.4.0) (judge verdict: Accepted)

mod = 998244353
eps = 10**-9


def main():
    import sys
    input = sys.stdin.readline

    # comb init
    # mod = 1000000007
    nmax = 25 * 10 ** 5 + 10  # change here
    fac = [0] * nmax
    finv = [0] * nmax
    inv = [0] * nmax
    fac[0] = 1
    fac[1] = 1
    finv[0] = 1
    finv[1] = 1
    inv[1] = 1
    for i in range(2, nmax):
        fac[i] = fac[i - 1] * i % mod
        inv[i] = mod - inv[mod % i] * (mod // i) % mod
        finv[i] = finv[i - 1] * inv[i] % mod

    def comb(n, r):
        if n < r:
            return 0
        else:
            return (fac[n] * ((finv[r] * finv[n - r]) % mod)) % mod

    N, M = map(int, input().split())
    ans = comb(3 * M + N - 1, N-1)
    for i in range(2 * M + 1, 3 * M+1):
        ans = (ans - (N * comb(3*M - i + N-2, N-2))%mod)%mod
    for i in range(M+2, 3 * M + 1, 2):
        if i > N:
            break
        j = (3 * M - i) // 2
        ans = (ans - (comb(N, i) * comb(j + N-1, N-1))%mod)%mod
    print(ans)


if __name__ == '__main__':
    main()