CF1731F Function Sum

Suppose you have an integer array $ a_1, a_2, \dots, a_n $ . Let $ \operatorname{lsl}(i) $ be the number of indices $ j $ ( $ 1 \le j < i $ ) such that $ a_j < a_i $ . Analogically, let $ \operatorname{grr}(i) $ be the number of indices $ j $ ( $ i < j \le n $ ) such that $ a_j > a_i $ . Let's name position $ i $ good in the array $ a $ if $ \operatorname{lsl}(i) < \operatorname{grr}(i) $ . Finally, let's define a function $ f $ on array $ a $ $ f(a) $ as the sum of all $ a_i $ such that $ i $ is good in $ a $ . Given two integers $ n $ and $ k $ , find the sum of $ f(a) $ over all arrays $ a $ of size $ n $ such that $ 1 \leq a_i \leq k $ for all $ 1 \leq i \leq n $ modulo $ 998\,244\,353 $ .

The first and only line contains two integers $ n $ and $ k $ ( $ 1 \leq n \leq 50 $ ; $ 2 \leq k < 998\,244\,353 $ ).

Output a single integer — the sum of $ f $ over all arrays $ a $ of size $ n $ modulo $ 998\,244\,353 $ .

In the first test case: $ f([1,1,1]) = 0 $ $ f([2,2,3]) = 2 + 2 = 4 $ $ f([1,1,2]) = 1 + 1 = 2 $ $ f([2,3,1]) = 2 $ $ f([1,1,3]) = 1 + 1 = 2 $ $ f([2,3,2]) = 2 $ $ f([1,2,1]) = 1 $ $ f([2,3,3]) = 2 $ $ f([1,2,2]) = 1 $ $ f([3,1,1]) = 0 $ $ f([1,2,3]) = 1 $ $ f([3,1,2]) = 1 $ $ f([1,3,1]) = 1 $ $ f([3,1,3]) = 1 $ $ f([1,3,2]) = 1 $ $ f([3,2,1]) = 0 $ $ f([1,3,3]) = 1 $ $ f([3,2,2]) = 0 $ $ f([2,1,1]) = 0 $ $ f([3,2,3]) = 2 $ $ f([2,1,2]) = 1 $ $ f([3,3,1]) = 0 $ $ f([2,1,3]) = 2 + 1 = 3 $ $ f([3,3,2]) = 0 $ $ f([2,2,1]) = 0 $ $ f([3,3,3]) = 0 $ $ f([2,2,2]) = 0 $ Adding up all of these values, we get $ 28 $ as the answer.