A15475 | Card Guessing

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.

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

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 $y \cdot y^{-1} \bmod 998244353 = 1$ ).