Problem:
Score : 1800 points

Problem Statement

We have an N \times M grid. The square at the i-th row and j-th column will be denoted as (i,j).
Particularly, the top-left square will be denoted as (1,1), and the bottom-right square will be denoted as (N,M).
Takahashi painted some of the squares (possibly zero) black, and painted the other squares white.

We will define an integer sequence A of length N, and two integer sequences B and C of length M each, as follows:

A_i(1\leq i\leq N) is the minimum j such that (i,j) is painted black, or M+1 if it does not exist.

B_i(1\leq i\leq M) is the minimum k such that (k,i) is painted black, or N+1 if it does not exist.

C_i(1\leq i\leq M) is the maximum k such that (k,i) is painted black, or 0 if it does not exist.

How many triples (A,B,C) can occur? Find the count modulo 998244353.

Notice

In this problem, the length of your source code must be at most 20000 B.
Note that we will invalidate submissions that exceed the maximum length.

Constraints

1 \leq N \leq 8000

1 \leq M \leq 200

N and M are integers.

Partial Score

1500 points will be awarded for passing the test set satisfying N\leq 300.

Input

Input is given from Standard Input in the following format:

N M

Output

Print the number of triples (A,B,C), modulo 998244353.

Sample Input 1

2 3

Sample Output 1

64

Since N=2, given B_i and C_i, we can uniquely determine the arrangement of black squares in each column.
For each i, there are four possible pairs (B_i,C_i): (1,1), (1,2), (2,2) and (3,0). Thus, the answer is 4^M=64.

Sample Input 2

4 3

Sample Output 2

2588

Sample Input 3

17 13

Sample Output 3

229876268

Sample Input 4

5000 100

Sample Output 4

57613837

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.*;
import java.math.*;
import java.util.*;

public class Main {

	ModuloCombinatorics mc = new ModuloCombinatorics(10000, P);

	static final int P = 998244353;

	static int nextPowerOf2(int x) {
		return x == 1 ? 1 : Integer.highestOneBit(x - 1) << 1;
	}

	static class CompV {
		double[] re, im;

		void add(CompV o) {
			for (int i = 0; i < re.length; i++) {
				re[i] += o.re[i];
				im[i] += o.im[i];
			}
		}

		static CompV times(CompV a, CompV b) {
			double[] aRe = a.re;
			double[] aIm = a.im;
			double[] bRe = b.re;
			double[] bIm = b.im;
			double[] re = new double[aRe.length];
			double[] im = new double[aRe.length];

			for (int i = 0; i < aRe.length; i++) {
				re[i] = aRe[i] * bRe[i] - aIm[i] * bIm[i];
				im[i] = aRe[i] * bIm[i] + aIm[i] * bRe[i];
			}

			return new CompV(re, im);
		}

		static CompV plus(CompV a, CompV b) {
			double[] aRe = a.re;
			double[] aIm = a.im;
			double[] bRe = b.re;
			double[] bIm = b.im;
			double[] re = new double[aRe.length];
			double[] im = new double[aRe.length];

			for (int i = 0; i < aRe.length; i++) {
				re[i] = aRe[i] + bRe[i];
				im[i] = aIm[i] + bIm[i];
			}

			return new CompV(re, im);
		}

		/**
		 * @return a + bi
		 */
		static CompV packFFT(CompV a, CompV b) {
			double[] aRe = a.re;
			double[] aIm = a.im;
			double[] bRe = b.re;
			double[] bIm = b.im;
			double[] re = new double[aRe.length];
			double[] im = new double[aRe.length];
			for (int i = 0; i < aRe.length; i++) {
				re[i] = aRe[i] - bIm[i];
				im[i] = aIm[i] + bRe[i];
			}
			return new CompV(re, im);
		}

		/**
		 * for real-valued a and b restores FFT(a) and FFT(b) from FFT(a + bi)
		 */
		void unpackFFT(CompV a, CompV b) {
			int n = re.length;
			double[] aRe = a.re;
			double[] aIm = a.im;
			double[] bRe = b.re;
			double[] bIm = b.im;
			for (int i = 0; i < n; i++) {
				int j = i == 0 ? 0 : n - i;
				aRe[i] = (re[i] + re[j]) * 0.5;
				aIm[i] = (im[i] - im[j]) * 0.5;
				bRe[i] = (im[i] + im[j]) * 0.5;
				bIm[i] = (re[j] - re[i]) * 0.5;
			}
		}

		public CompV(double[] re, double[] im) {
			this.re = re;
			this.im = im;
		}

		public CompV(int len) {
			re = new double[len];
			im = new double[len];
		}
	}

	static final int L = (int) Math.sqrt(P);
	static final int L2 = L * L;

	public static int[] multFaster(int[] a, int[] b) {
		int len = nextPowerOf2(a.length + b.length - 1);

		double[] p1 = new double[len];
		double[] q1 = new double[len];
		for (int i = 0; i < a.length; i++) {
			p1[i] = a[i] % L;
			q1[i] = a[i] / L;
		}

		double[] p2 = new double[len];
		double[] q2 = new double[len];
		for (int i = 0; i < b.length; i++) {
			p2[i] = b[i] % L;
			q2[i] = b[i] / L;
		}

		CompV p1p2Pack = new CompV(p1, p2);
		CompV q1q2Pack = new CompV(q1, q2);

		fft(p1p2Pack, false);
		fft(q1q2Pack, false);

		CompV fp1 = new CompV(len);
		CompV fp2 = new CompV(len);
		p1p2Pack.unpackFFT(fp1, fp2);

		CompV fq1 = new CompV(len);
		CompV fq2 = new CompV(len);
		q1q2Pack.unpackFFT(fq1, fq2);

		CompV back0 = CompV.times(fp1, fp2);

		CompV back1 = CompV.times(fp1, fq2);
		back1.add(CompV.times(fq1, fp2));

		CompV back2 = CompV.times(fq1, fq2);

		CompV back01 = CompV.packFFT(back0, back1);

		fft(back01, true);
		fft(back2, true);

		int[] ret = new int[len];

		for (int i = 0; i < len; i++) {
			long v0 = Math.round(back01.re[i]) % P;
			long v1 = Math.round(back01.im[i]) % P * L % P;
			long v2 = Math.round(back2.re[i]) % P * L2 % P;
			ret[i] = (int) ((v0 + v1 + v2) % P);
		}

		return ret;
	}

	public static void fft(CompV v, boolean invert) {
		double[] a = v.re;
		double[] b = v.im;
		int n = a.length;
		int shift = 32 - Integer.numberOfTrailingZeros(n);
		prepareArrays(32 - shift);
		int[] rev = rev2D[32 - shift];
		// System.err.println(n + " " + shift);
		for (int i = 1; i < n; i++) {
			int j = rev[i];
			if (i < j) {
				double temp = a[i];
				a[i] = a[j];
				a[j] = temp;
				temp = b[i];
				b[i] = b[j];
				b[j] = temp;
			}
		}

		// this shit looks awkward
		double multInv = invert ? -1 : 1;

		for (int len = 2, row = 0; len <= n; len <<= 1, row++) {
			int halfLen = len >> 1;
			double[] multReArr = powRe[row];
			double[] multImArr = powIm[row];
			for (int i = 0; i < n; i += len) {
				int toJ = i + halfLen;

				for (int j1 = i, j2 = toJ, k = 0; k < halfLen; j1++, j2++, k++) {
					double uA = a[j1];
					double uB = b[j1];

					double multRe = multReArr[k];
					double multIm = multImArr[k] * multInv;

					double vA = a[j2] * multRe - b[j2] * multIm;
					double vB = a[j2] * multIm + b[j2] * multRe;
					a[j1] = uA + vA;
					b[j1] = uB + vB;
					a[j2] = uA - vA;
					b[j2] = uB - vB;
				}
			}
		}
		if (invert) {
			for (int i = 0; i < n; i++) {
				a[i] /= n;
				b[i] /= n;
			}
		}
	}

	static double[][] powRe = { { 1 } };
	static double[][] powIm = { { 0 } };
	static int[][] rev2D = {};

	static void prepareArrays(int n) {

		if (rev2D.length < n + 1) {
			rev2D = Arrays.copyOf(rev2D, n + 1);
		}

		if (rev2D[n] == null) {
			int[] tmp = rev2D[n] = new int[1 << n];
			for (int i = 0; i < (1 << n); i++) {
				tmp[i] = tmp[i >> 1] >> 1;
				if ((i & 1) == 1) {
					tmp[i] |= 1 << (n - 1);
				}
			}
		}

		int oldN = powRe.length;

		if (oldN >= n) {
			return;
		}

		powRe = Arrays.copyOf(powRe, n);
		powIm = Arrays.copyOf(powIm, n);

		for (int i = oldN; i < n; i++) {

			double angle = Math.PI / (1 << i);

			double multRe = Math.cos(angle);
			double multIm = Math.sin(angle);

			double[] toRe = powRe[i] = new double[1 << i];
			double[] toIm = powIm[i] = new double[1 << i];

			double[] fromRe = powRe[i - 1];
			double[] fromIm = powIm[i - 1];

			for (int j = 0; j < 1 << (i - 1); j++) {

				double re = fromRe[j];
				double im = fromIm[j];

				toRe[j << 1] = re;
				toIm[j << 1] = im;
				toRe[(j << 1) | 1] = re * multRe - im * multIm;
				toIm[(j << 1) | 1] = re * multIm + im * multRe;
			}

			// System.err.println(Arrays.toString(toRe));
			// System.err.println(Arrays.toString(toIm));
		}
	}

	void submit() {
		int r = nextInt();
		int c = nextInt();
		int[] dp = new int[r + 1];
		Arrays.fill(dp, 1);
		c--;

		int[] invFact = mc.invFact;
		int[] fact = mc.fact;
		
		int[] two = new int[r + 1];
		for (int i = 1; i <= r; i++) {
			two[i] = invFact[i + 2];
		}

		while (c-- > 0) {
			int[] tmp = new int[r + 1];
			for (int i = 0; i <= r; i++) {
				tmp[i] = (int)((long)dp[i] * invFact[i] % P);
			}
			int[] next = multFaster(tmp, two);
			next = Arrays.copyOf(next, r + 1);
			for (int i = 0; i <= r; i++) {
				next[i] = (int)((long)next[i] * fact[i + 2] % P);
			}
			for (int i = 0; i <= r; i++) {
				int ways = 1 + (i + 1) * i / 2;
				next[i] += (int) ((long) dp[i] * ways % P);
				if (next[i] >= P) {
					next[i] -= P;
				}
			}
			dp = next;
		}

		int ret = 0;
		for (int i = 0; i <= r; i++) {
			ret += (int) ((long) mc.choose(r, i) * dp[i] % P);
			if (ret >= P) {
				ret -= P;
			}
		}
		
		out.println(ret);
	}

	static class ModuloCombinatorics {
		/** maximal needed number, N itself is included **/
		final int N;

		/** prime modulo **/
		final int P;

		/** factorials **/
		final int[] fact;

		/** multiplicative inverses, take care to not touch inv[0] **/
		final int[] inv;

		/** inverse factorials **/
		final int[] invFact;

		public ModuloCombinatorics(int N, int P) {
			this.N = N;
			this.P = P;
			fact = new int[N + 1];
			fact[0] = 1;
			for (int i = 1; i <= N; i++) {
				fact[i] = (int) ((long) i * fact[i - 1] % P);
			}

			inv = new int[N + 1];
			inv[1] = 1;
			for (int i = 2; i <= N; i++) {
				inv[i] = P - (int) ((long) (P / i) * inv[P % i] % P);
			}

			invFact = new int[N + 1];
			invFact[0] = 1;
			for (int i = 1; i <= N; i++) {
				invFact[i] = (int) ((long) invFact[i - 1] * inv[i] % P);
			}
		}

		public int choose(int n, int k) {
			return (n < 0 || k < 0 || k > n) ? 0 : (int) ((long) fact[n]
					* invFact[k] % P * invFact[n - k] % P);
		}

		/** a^b modulo mod, mod is arbitrary **/
		static public int pow(int a, long b, int mod) {
			if (a < 0 || a >= mod || b < 0) {
				throw new IllegalArgumentException();
			}
			int ret = 1;
			for (; b > 0; b >>= 1) {
				if ((b & 1) == 1) {
					ret = (int) ((long) ret * a % mod);
				}
				a = (int) ((long) a * a % mod);
			}
			return ret;
		}

		/** a^b modulo P **/
		public int pow(int a, long b) {	
			return pow(a, b, P);
		}
	}

	void preCalc() {

	}

	void test() {

	}

	void stress() {
		for (int tst = 0;; tst++) {
			if (false) {
				throw new AssertionError();
			}
			System.err.println(tst);
		}
	}

	Main() throws IOException {
		is = System.in;
		out = new PrintWriter(System.out);
		preCalc();
		submit();
		// stress();
		// test();
		out.close();
	}

	static final Random rng = new Random();
	static final int C = 5;

	static int rand(int l, int r) {
		return l + rng.nextInt(r - l + 1);
	}

	public static void main(String[] args) throws IOException {
		new Main();
	}

	private InputStream is;
	PrintWriter out;

	private byte[] buf = new byte[1 << 14];
	private int bufSz = 0, bufPtr = 0;

	private int readByte() {
		if (bufSz == -1)
			throw new RuntimeException("Reading past EOF");
		if (bufPtr >= bufSz) {
			bufPtr = 0;
			try {
				bufSz = is.read(buf);
			} catch (IOException e) {
				throw new RuntimeException(e);
			}
			if (bufSz <= 0)
				return -1;
		}
		return buf[bufPtr++];
	}

	private boolean isTrash(int c) {
		return c < 33 || c > 126;
	}

	private int skip() {
		int b;
		while ((b = readByte()) != -1 && isTrash(b))
			;
		return b;
	}

	String nextToken() {
		int b = skip();
		StringBuilder sb = new StringBuilder();
		while (!isTrash(b)) {
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}

	String nextString() {
		int b = readByte();
		StringBuilder sb = new StringBuilder();
		while (!isTrash(b) || b == ' ') {
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}

	double nextDouble() {
		return Double.parseDouble(nextToken());
	}

	char nextChar() {
		return (char) skip();
	}

	int nextInt() {
		int ret = 0;
		int b = skip();
		if (b != '-' && (b < '0' || b > '9')) {
			throw new InputMismatchException();
		}
		boolean neg = false;
		if (b == '-') {
			neg = true;
			b = readByte();
		}
		while (true) {
			if (b >= '0' && b <= '9') {
				ret = ret * 10 + (b - '0');
			} else {
				if (b != -1 && !isTrash(b)) {
					throw new InputMismatchException();
				}
				return neg ? -ret : ret;
			}
			b = readByte();
		}
	}

	long nextLong() {
		long ret = 0;
		int b = skip();
		if (b != '-' && (b < '0' || b > '9')) {
			throw new InputMismatchException();
		}
		boolean neg = false;
		if (b == '-') {
			neg = true;
			b = readByte();
		}
		while (true) {
			if (b >= '0' && b <= '9') {
				ret = ret * 10 + (b - '0');
			} else {
				if (b != -1 && !isTrash(b)) {
					throw new InputMismatchException();
				}
				return neg ? -ret : ret;
			}
			b = readByte();
		}
	}
}