Serval and Bonus Problem
Description
Getting closer and closer to a mathematician, Serval becomes a university student on math major in Japari University. On the Calculus class, his teacher taught him how to calculate the expected length of a random subsegment of a given segment. Then he left a bonus problem as homework, with the award of a garage kit from IOI. The bonus is to extend this problem to the general case as follows.
You are given a segment with length $l$. We randomly choose $n$ segments by choosing two points (maybe with non-integer coordinates) from the given segment equiprobably and the interval between the two points forms a segment. You are given the number of random segments $n$, and another integer $k$. The $2n$ endpoints of the chosen segments split the segment into $(2n+1)$ intervals. Your task is to calculate the expected total length of those intervals that are covered by at least $k$ segments of the $n$ random segments.
You should find the answer modulo $998244353$.
First line contains three space-separated positive integers $n$, $k$ and $l$ ($1\leq k \leq n \leq 2000$, $1\leq l\leq 10^9$).
Output one integer — the expected total length of all the intervals covered by at least $k$ segments of the $n$ random segments modulo $998244353$.
Formally, let $M = 998244353$. It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$. Output the integer equal to $p \cdot q^{-1} \bmod M$. In other words, output such an integer $x$ that $0 \le x < M$ and $x \cdot q \equiv p \pmod{M}$.
Input
First line contains three space-separated positive integers $n$, $k$ and $l$ ($1\leq k \leq n \leq 2000$, $1\leq l\leq 10^9$).
Output
Output one integer — the expected total length of all the intervals covered by at least $k$ segments of the $n$ random segments modulo $998244353$.
Formally, let $M = 998244353$. It can be shown that the answer can be expressed as an irreducible fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$. Output the integer equal to $p \cdot q^{-1} \bmod M$. In other words, output such an integer $x$ that $0 \le x < M$ and $x \cdot q \equiv p \pmod{M}$.
Samples
1 1 1
332748118
6 2 1
760234711
7 5 3
223383352
97 31 9984524
267137618
Note
In the first example, the expected total length is $\int_0^1 \int_0^1 |x-y| \,\mathrm{d}x\,\mathrm{d}y = {1\over 3}$, and $3^{-1}$ modulo $998244353$ is $332748118$.