407979: GYM102956 G Biological Software Utilities

You are developing a software kit named Biological Software Utilities (BSU). The kit includes a program that is dedicated to tree recognition. Recall that a tree is a connected undirected graph without cycles.

In nature, when a tree grows, two neighboring vertices are added at the same time. Thus, you consider a tree to be plausible if, after removing some edges, the resulting graph consists only of connected components with $$$2$$$ vertices. In other words, a tree is plausible if and only if it has a perfect matching.

Now you are to implement a new function for BSU to calculate the number of plausible trees that have $$$n$$$ vertices numbered with distinct integers between $$$1$$$ and $$$n$$$. Two trees are considered different if there is an edge $$$(u, v)$$$ which is present in exactly one of the trees.

Since the number of plausible trees can be very large, you have to calculate it modulo $$$998\,244\,353$$$.

The only line contains an integer $$$n$$$, the number of vertices in a tree ($$$1 \le n \le 10^6$$$).

Print the number of plausible trees with $$$n$$$ vertices modulo $$$998\,244\,353$$$.

1

0

2

1

3

0

4

12

7788

178152092