Card Guessing

题意翻译

有 $4$ 种花⾊的牌，每种牌均为 $n$ 张，则牌的排列⼀共有 $(4\cdot n)!$ 种。 现在你从牌堆中逐张地取出牌，取牌之前你都会猜⼀下这张牌是什么花⾊。你会根据之前的 $k$ 张牌中出现最少的花⾊来猜这张牌。 如果有多种花⾊都是最少的，你随机地猜⼀种。如果之前抽出的牌不⾜ $k$ 张，就按之前的所有牌中的最少花⾊来猜。 问你猜对的期望次数是多少？

题目描述

Consider a deck of cards. Each card has one of $ 4 $ suits, and there are exactly $ n $ cards for each suit — so, the total number of cards in the deck is $ 4n $ . The deck is shuffled randomly so that each of $ (4n)! $ possible orders of cards in the deck has the same probability of being the result of shuffling. Let $ c_i $ be the $ i $ -th card of the deck (from top to bottom). Monocarp starts drawing the cards from the deck one by one. Before drawing a card, he tries to guess its suit. Monocarp remembers the suits of the $ k $ last cards, and his guess is the suit that appeared the least often among the last $ k $ cards he has drawn. So, while drawing the $ i $ -th card, Monocarp guesses that its suit is the suit that appears the minimum number of times among the cards $ c_{i-k}, c_{i-k+1}, \dots, c_{i-1} $ (if $ i \le k $ , Monocarp considers all previously drawn cards, that is, the cards $ c_1, c_2, \dots, c_{i-1} $ ). If there are multiple suits that appeared the minimum number of times among the previous cards Monocarp remembers, he chooses a random suit out of those for his guess (all suits that appeared the minimum number of times have the same probability of being chosen). After making a guess, Monocarp draws a card and compares its suit to his guess. If they match, then his guess was correct; otherwise it was incorrect. Your task is to calculate the expected number of correct guesses Monocarp makes after drawing all $ 4n $ cards from the deck.

输入输出格式

输入格式

The first (and only) line contains two integers $ n $ ( $ 1 \le n \le 500 $ ) and $ k $ ( $ 1 \le k \le 4 \cdot n $ ).

输出格式

Let the expected value you have to calculate be an irreducible fraction $ \dfrac{x}{y} $ . Print one integer — the value of $ x \cdot y^{-1} \bmod 998244353 $ , where $ y^{-1} $ is the inverse to $ y $ (i. e. an integer such that $ y \cdot y^{-1} \bmod 998244353 = 1 $ ).

输入输出样例

输入样例 #1

1 1

输出样例 #1

748683266

输入样例 #2

3 2

输出样例 #2

567184295

输入样例 #3

2 7

输出样例 #3

373153250

输入样例 #4

2 8

输出样例 #4

373153250