Redundant Binary Notation

Binomial trees of a binomial heap. Wikimedia, cc-by-sa

Redundant binary notation is similar to binary notation, except instead of allowing only $0$’s and $1$’s for each digit, we allow any integer digit in the range $\[0, t]$, where $t$ is some specified upper bound. For example, if $t = 2$, the digit $2$ is permitted, and we may write the decimal number $4$ as $100$, $20$, or $12$. If $t=1$, every number has precisely one representation, which is its typical binary representation. In general, if a number is written as $d\_l d\_{l-1} \ldots d\_1 d\_0$ in redundant binary notation, the equivalent decimal number is $d\_l\cdot2^l + d\_{l-1}\cdot2^{l-1} + \cdots + d\_1\cdot2^1 + d\_0\cdot2^0$.

Redundant binary notation can allow carryless arithmetic, and thus has applications in hardware design and even in the design of worst-case data structures. For example, consider insertion into a standard binomial heap. This operation takes $O(\log n)$ worst-case time but $O(1)$ amortized time. This is because the binary number representing the total number of elements in the heap can be incremented in $O(\log n)$ worst-case time and $O(1)$ amortized time. By using a redundant binary representation of the individual binomial trees in a binomial heap, it is possible to improve the worst-case insertion time of binomial heaps to $O(1)$.

However, none of that information is relevant to this question. In this question, your task is simple. Given a decimal number $N$ and the digit upper bound $t$, you are to count the number of possible representations $N$ has in redundant binary notation with each digit in range $\[0, t]$ with no leading zeros.

Input consists of a single line with two decimal integers $N$ ($0 \leq N \leq 10^{16}$) and $t$ ($1 \leq t \leq 100$).

Output in decimal the number of representations the decimal number $N$ has in redundant binary notation with each digit in range $\[0, t]$ with no leading zeros. Since the number of representations may be very large, output the answer modulo the large prime $998\\,244\\,353$.