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 JavaScript (node.js v5.12) (judge verdict: Accepted)

var GET=(function(){function f(s){return new g(s);}function g(s){this._s=s.trim().split("\n");this._y=0;}g.prototype.a=function(f){var s=this._s, y=this._y, r;if(typeof s[y]==="string")s[y]=s[y].split(" ").reverse();r=s[y].pop();if(!s[y].length)this._y++;return f?r:+r;};g.prototype.l=function(f){var s=this._s[this._y++].split(" ");return f?s:s.map(a=>+a);};g.prototype.m=function(n,f){var r=this._s.slice(this._y,this._y+=n).map(a=>a.split(" "));return f?r:r.map(a=>a.map(a=>+a));};g.prototype.r=function(n,f){var r=this._s.slice(this._y,this._y+=n);return f?r:r.map(a=>+a);};return f;})();
var o=GET(require("fs").readFileSync("/dev/stdin","utf8"));
function xArray(v){var a=arguments,l=a.length,r="Array(a["+--l+"]).fill(0).map(x=>{return "+v+";})";while(--l)r="Array(a["+l+"]).fill(0).map(x=>"+r+")";return eval(r);}
var mod = 998244353;
function abmod(){for(var a=arguments,r=a[0],i=1;i<a.length;)r=((r>>16)*a[i]%mod*65536+(r&65535)*a[i++])%mod;return r;}
var fac=[1,1];
function fSet(n){for(var i=2;i<=n;)fac[i]=abmod(fac[i-1],i++);}
function modinv(a){for(var b=mod,u=1,v=0,t;b;v=t)t=a/b|0,a-=t*b,u-=t*v,t=a,a=b,b=t,t=u,u=v;u%=mod;return u<0?u+mod:u;}
function nCrmod(n,r){return abmod(fac[n],modinv(abmod(fac[r],fac[n-r])));}

console.log(main());
function main(){
  var n = o.a();
  var m = o.a();
  var m3 = 3*m;
  fSet(m3+n);
  var ans = 0;
  var mx = Math.min(n,m);
  for(var i = m&1; i <= mx; i += 2){
    var j = m3-i >> 1;
    var t = nCrmod(j+n-1,j);
    t = abmod(t, nCrmod(n, i))
    ans += t;
  }
  for(var i = 0; i < m; i++){
    var t = abmod(n, nCrmod(i+n-2,i));
    ans -= t;
  }
  ans %= mod;
  ans += mod;
  return ans % mod;
}