A14742 | 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 $998\,244\,353$ .