Problem:
Score : 700 points

Problem Statement

Takahashi throws N dice, each having K sides with all integers from 1 to K. The dice are NOT pairwise distinguishable.
For each i=2,3,...,2K, find the following value modulo 998244353:

The number of combinations of N sides shown by the dice such that the sum of no two different sides is i.

Note that the dice are NOT distinguishable, that is, two combinations are considered different when there exists an integer k such that the number of dice showing k is different in those two.

Constraints

1 \leq K \leq 2000

2 \leq N \leq 2000

K and N are integers.

Input

Input is given from Standard Input in the following format:

K N

Output

Print 2K-1 integers. The t-th of them (1\leq t\leq 2K-1) should be the answer for i=t+1.

Sample Input 1

3 3

Sample Output 1

7
7
4
7
7

For i=2, the combinations (1,2,2),(1,2,3),(1,3,3),(2,2,2),(2,2,3),(2,3,3),(3,3,3) satisfy the condition, so the answer is 7.

For i=3, the combinations (1,1,1),(1,1,3),(1,3,3),(2,2,2),(2,2,3),(2,3,3),(3,3,3) satisfy the condition, so the answer is 7.

For i=4, the combinations (1,1,1),(1,1,2),(2,3,3),(3,3,3) satisfy the condition, so the answer is 4.

Sample Input 2

4 5

Sample Output 2

36
36
20
20
20
36
36

Sample Input 3

6 1000

Sample Output 3

149393349
149393349
668669001
668669001
4000002
4000002
4000002
668669001
668669001
149393349
149393349

Solution in Java (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.io.*;
import java.util.*;
public class Main{
	static int mod=998244353;
	static long comb[][];
	static void nCr2(int N)		// O(N * N)
	{
		comb = new long[N][N];
		comb[0][0] = 1;
		for (int i = 1; i < N; i++)
		{
			comb[i][0] = 1;
			for (int j = 1; j <= i; j++)
				comb[i][j] = ((comb[i-1][j] + comb[i-1][j-1]))%mod;		//may use mod
		}
	}
	static long count(int dices,int faces) {
		return comb[dices+faces-1][faces-1];
	}
	static void main() throws Exception{
		nCr2(4040);
		int k=sc.nextInt(),n=sc.nextInt();
		long[]pairs=new long[4001];
		for(int i=0;i<=n/2;i++) {
			pairs[i]=count(n-i*2, k);
		}
		
		for(int i=2;i<=2*k;i++) {
			int curPair=Math.min(i, k+1)-((i+1)/2);
			long ans=0;
			for(int p=0;p<=curPair;p++) {
				if((p&1)==0) {
					long inc=(comb[curPair][p]*pairs[p])%mod;
					ans=(ans+inc)%mod;
				}
				else {
					long dec=(comb[curPair][p]*pairs[p])%mod;
					ans=(ans-dec)%mod;
				}
			}
			pw.println(ans);
		}
	}
	public static void main(String[] args) throws Exception{
		sc=new MScanner(System.in);
		pw = new PrintWriter(System.out);
		int tc=1;
//		tc=sc.nextInt();
		while(tc-->0)
			main();
		pw.flush();
	}
	static PrintWriter pw;
	static MScanner  sc;
	static class MScanner {
		StringTokenizer st;
		BufferedReader br;
		public MScanner(InputStream system) {
			br = new BufferedReader(new InputStreamReader(system));
		}
 
		public MScanner(String file) throws Exception {
			br = new BufferedReader(new FileReader(file));
		}
 
		public String next() throws IOException {
			while (st == null || !st.hasMoreTokens())
				st = new StringTokenizer(br.readLine());
			return st.nextToken();
		}
		public int[] intArr(int n) throws IOException {
	        int[]in=new int[n];for(int i=0;i<n;i++)in[i]=nextInt();
	        return in;
		}
		public long[] longArr(int n) throws IOException {
	        long[]in=new long[n];for(int i=0;i<n;i++)in[i]=nextLong();
	        return in;
		}
		public int[] intSortedArr(int n) throws IOException {
	        int[]in=new int[n];for(int i=0;i<n;i++)in[i]=nextInt();
	        shuffle(in);
	        Arrays.sort(in);
	        return in;
		}
		public long[] longSortedArr(int n) throws IOException {
	        long[]in=new long[n];for(int i=0;i<n;i++)in[i]=nextLong();
	        shuffle(in);
	        Arrays.sort(in);
	        return in;
		}
		public Integer[] IntegerArr(int n) throws IOException {
	        Integer[]in=new Integer[n];for(int i=0;i<n;i++)in[i]=nextInt();
	        return in;
		}
		public Long[] LongArr(int n) throws IOException {
	        Long[]in=new Long[n];for(int i=0;i<n;i++)in[i]=nextLong();
	        return in;
		}
		public String nextLine() throws IOException {
			return br.readLine();
		}
 
		public int nextInt() throws IOException {
			return Integer.parseInt(next());
		}
 
		public double nextDouble() throws IOException {
			return Double.parseDouble(next());
		}
 
		public char nextChar() throws IOException {
			return next().charAt(0);
		}
 
		public long nextLong() throws IOException {
			return Long.parseLong(next());
		}
 
		public boolean ready() throws IOException {
			return br.ready();
		}
 
		public void waitForInput() throws InterruptedException {
			Thread.sleep(3000);
		}
	}
	static void shuffle(int[]in) {
		for(int i=0;i<in.length;i++) {
			int idx=(int)(Math.random()*in.length);
			int tmp=in[i];
			in[i]=in[idx];
			in[idx]=tmp;
		}
	}
	static void shuffle(long[]in) {
		for(int i=0;i<in.length;i++) {
			int idx=(int)(Math.random()*in.length);
			long tmp=in[i];
			in[i]=in[idx];
			in[idx]=tmp;
		}
	}
}