A15460. 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.