Sum of Product of Binomial Coefficients

You are given integers $N$ and $K$. For a positive integer $k$, $f(k)$ is defined as follows.

• The Sum of $\binom{N}{a\_1} \times \binom{a\_1}{a\_2} \times \cdots \times \binom{a\_{k-1}}{a\_k}$ for all integer sequences $(a\_1, a\_2, \dots, a\_k)$ that satisfy the condition $N \ge a\_1 \ge a\_2 \ge \dots \ge a\_k \ge 0$.

Answer the remainder of $\sum\_{k=1}^{K}{f(k)}$ divided by $998244353$.

For each input, solve $T$ test cases.

Note that $\binom{A}{B}$ represents "the number of ways to select $B$ distinct items from $A$ items" (i.e., the binomial coefficient).

$T$

$\text{case}\_1$

$\vdots$

$\text{case}\_T$

Each test case is given in the following format.

$N$ $K$

Output the remainder of $\sum\_{k=1}^{K}{f(k)}$ divided by $998244353$ for each test case.

• All test cases consist of integers.
• $1 \le T \le 10^5$
• $0 \le N \le 10^9$
• $1 \le K \le 2 \times 10^5$
• The sum of $K$ in one test case does not exceed $2 \times 10^5$.