CF2278F Modulo and Counting

You are given two integers $ n $ and $ m $ . An array $ a $ of length $ n $ is called good if it satisfies the following two conditions: - $ \color{red}{m + 2} \ge a_1 \ge a_2 \ge \ldots \ge a_{n-1} \ge a_n \ge 1 $ ; - $ (a_1 \bmod a_2) + (a_2 \bmod a_3) + \ldots + (a_{n-1} \bmod a_n) = m $ . Your task is to find the number of good arrays. Since the answer can be very large, report it modulo $ 998\,244\,353 $ .

The first line of the input contains a single integer $ t $ ( $ 1 \le t \le 2\,000 $ ) — the number of test cases. The description of each test case follows. The only line of each test case contains two space-separated integers $ n $ and $ m $ ( $ 2 \le n \le 4\,000 $ ; $ 1 \le m \le 4\,000 $ ). It is guaranteed that the sum of $ n $ over all the test cases does not exceed $ 4000 $ and the sum of $ m $ over all the test cases does not exceed $ 4000 $ .

For each test case, print the number of good arrays. As the answer can be large, output it modulo $ 998\,244\,353 $ .

For the first test case, the good arrays are: - $ [3, 2, 1] $ : $ (3 \bmod 2) + (2 \bmod 1) = 1 + 0 = 1 $ . - $ [3, 2, 2] $ : $ (3 \bmod 2) + (2 \bmod 2) = 1 + 0 = 1 $ . - $ [3, 3, 2] $ : $ (3 \bmod 3) + (3 \bmod 2) = 0 + 1 = 1 $ . For the second test case, the only good array is $ [4, 3, 2] $ .