A16379. Counting Binary Strings

Patrick calls a substring $^\dagger$ of a binary string $^\ddagger$ good if this substring contains exactly one 1.

Help Patrick count the number of binary strings $s$ such that $s$ contains exactly $n$ good substrings and has no good substring of length strictly greater than $k$ . Note that substrings are differentiated by their location in the string, so if $s =$ 1010 you should count both occurrences of 10.

$^\dagger$ A string $a$ is a substring of a string $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) characters from the beginning and several (possibly, zero or all) characters from the end.

$^\ddagger$ A binary string is a string that only contains the characters 0 and 1.

Each test consists of multiple test cases. The first line contains a single integer $t$ ( $1 \leq t \leq 2500$ ) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $n$ and $k$ ( $1 \leq n \leq 2500$ , $1 \leq k \leq n$ ) — the number of required good substrings and the maximum allowed length of a good substring.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2500$ .

For each test case, output a single integer — the number of binary strings $s$ such that $s$ contains exactly $n$ good substrings and has no good substring of length strictly greater than $k$ . Since this integer can be too large, output it modulo $998\,244\,353$ .

In the first test case, the only suitable binary string is 1. String 01 is not suitable because it contains a substring 01 with length $2 > 1$ .

In the second test case, suitable binary strings are 011, 110 and 111.

In the third test case, suitable binary strings are 101, 0110, 0111, 1110, and 1111.