P5223 Function

${\rm CYJian}$ recently thought of [Water Triangle](https://www.luogu.org/problemnew/show/P5014) and felt it was too easy, so he came up with a more interesting version.

Given $N$ and $K$, compute: $$\sum_{i=1}^{K} f[N][i] \ (\bmod\ 998244353)$$ where: $$f[i][j] = f[i-1][j] + f[i][j-1] + f[i-1][j-1] \ (i>1, j \leq i)$$ $$f[1][1] = 1 \qquad f[i][0] = 0 \qquad f[i][j] = 0 \ (j>i)$$

The first line contains two positive integers $N$ and $K$.

Output one line containing the value of the expression above.

Constraints: For $10\%$ of the testdata: $1 \leq N \leq 10^3 \qquad 1 \leq K \leq 10^2$. For $30\%$ of the testdata: $1 \leq N \leq 10^6 \qquad 1 \leq K \leq 10^2$. For $50\%$ of the testdata: $1 \leq N \leq 10^{18} \qquad 1 \leq K \leq 10^2$. For another $20\%$ of the testdata: $1 \leq N \leq 10^6 \qquad 1 \leq K \leq 10^3$. For $100\%$ of the testdata: $1 \leq N \leq 10^{18} \qquad 1 \leq K \leq 10^3$. It is guaranteed that $K \leq N$. Update: The time limit has been changed as follows: for test points $1$~$35$, the time limit is $600 \text{ ms}$; for test points $36$~$50$, the time limit is $400 \text{ ms}$. Translated by ChatGPT 5