Function Sum
Description
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 $998\,244\,353$.
Input
The first and only line contains two integers $n$ and $k$ ($1 \leq n \leq 50$; $2 \leq k < 998\,244\,353$).
Output
Output a single integer — the sum of $f$ over all arrays $a$ of size $n$ modulo $998\,244\,353$.
3 3
5 6
12 30
28
34475
920711694
Note
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.