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 $998244353$.

2 2

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 $(3,3)$.

3 2

19

10 10

211428932

100000 50000

3463133