CF1821F Timber

There is a beautiful alley with trees in front of a shopping mall. Unfortunately, it has to go to make space for the parking lot. The trees on the alley all grow in a single line. There are $ n $ spots for trees, index $ 0 $ is the shopping mall, index $ n+1 $ is the road and indices from $ 1 $ to $ n $ are the spots for trees. Some of them are taken — there grow trees of the same height $ k $ . No more than one tree grows in each spot. When you chop down a tree in the spot $ x $ , you can make it fall either left or right. If it falls to the left, it takes up spots from $ x-k $ to $ x $ , inclusive. If it falls to the right, it takes up spots from $ x $ to $ x+k $ , inclusive. Let $ m $ trees on the alley grow in some spots $ x_1, x_2, \dots, x_m $ . Let an alley be called unfortunate if all $ m $ trees can be chopped down in such a way that: - no tree falls on the shopping mall or the road; - each spot is taken up by no more than one fallen tree. Calculate the number of different unfortunate alleys with $ m $ trees of height $ k $ . Two alleys are considered different if there is a spot $ y $ such that a tree grows in $ y $ on the first alley and doesn't grow in $ y $ on the second alley. Output the number modulo $ 998\,244\,353 $ .

The only line contains three integers $ n, m $ and $ k $ ( $ 1 \le m, k \le n \le 3 \cdot 10^5 $ ) — the number of spots for the trees, the number of trees and the height of each tree.

Print a single integer — the number of different unfortunate alleys with $ m $ trees of height $ k $ , modulo $ 998\,244\,353 $ .