Score : $1100$ points

Problem Statement

We have a bar of length $1$. A point on the bar whose distance from the left end of the bar is $x$ is said to have a coordinate $x$.

Snuke will do the operation below $N$ times.

• Choose two real numbers $x$ and $y$ uniformly at random from $[0, 1]$. Put a sticker covering the range from the coordinate $\min(x,y)$ to the coordinate $\max(x,y)$.

Here, all random choices are independent of each other.

Stickers can overlap. The bar is said to be good when no point is covered by $K+1$ or more stickers.

Find the probability, modulo $998244353$, of having a good bar after putting $N$ stickers.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq K \leq \min(N,10^5)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $K$

Output

Print the answer.

2 1

332748118

We are to find the probability that the two stickers do not overlap, which is $1/3$.

5 3

66549624

10000 5000

642557092