CF1608F MEX counting

For an array $ c $ of nonnegative integers, $ MEX(c) $ denotes the smallest nonnegative integer that doesn't appear in it. For example, $ MEX([0, 1, 3]) = 2 $ , $ MEX([42]) = 0 $ . You are given integers $ n, k $ , and an array $ [b_1, b_2, \ldots, b_n] $ . Find the number of arrays $ [a_1, a_2, \ldots, a_n] $ , for which the following conditions hold: - $ 0 \le a_i \le n $ for each $ i $ for each $ i $ from $ 1 $ to $ n $ . - $ |MEX([a_1, a_2, \ldots, a_i]) - b_i| \le k $ for each $ i $ from $ 1 $ to $ n $ . As this number can be very big, output it modulo $ 998\,244\,353 $ .

The first line of the input contains two integers $ n, k $ ( $ 1 \le n \le 2000 $ , $ 0 \le k \le 50 $ ). The second line of the input contains $ n $ integers $ b_1, b_2, \ldots, b_n $ ( $ -k \le b_i \le n+k $ ) — elements of the array $ b $ .

Output a single integer — the number of arrays which satisfy the conditions from the statement, modulo $ 998\,244\,353 $ .