Serval and Bonus Problem

有一段长为$l$的线段，有$n$个区间，左右端点在$[0,l)$间均匀随机（可能不是整数） 求期望被至少$k$段区间覆盖的长度，对998244353取膜

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} $ .

1 1 1

332748118

6 2 1

760234711

7 5 3

223383352

97 31 9984524

267137618

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 $ .