Problem Statement

We have a rooted tree T with N vertices numbered 1 to N. Vertex 1 is the root, and the parent of vertex $i (2 \leq i \leq N)$ is vertex P_i.
A non-empty subset S of the vertex set $V = \lbrace 1, 2,\dots, N\rbrace$ of T is said to be a good vertex set when it satisfies the following condition.

• For every pair of different vertices (u, v) in S, the following holds: u is not an ancestor of v.

For each $K = 1, 2, \dots, N,$ find the number, modulo 998244353, of good vertex sets with exactly K vertices.

Input

The input is given from Standard Input in the following format:
N
$P_2 P_3 \dots P_N$

Output

Print N lines. The i-th line should contain the answer for K = i.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq P_i \lt i$
• All values in the input are integers.

Sample Input 1

4
1 2 1

Sample Output 1

4
2
0
0

For each $1 \leq K \leq N$, the good vertex sets of size K are listed below.

• K=1: $\lbrace 1 \rbrace, \lbrace 2 \rbrace, \lbrace 3 \rbrace, \lbrace 4 \rbrace$.
• K=2: $\lbrace 2, 4 \rbrace, \lbrace 3, 4 \rbrace$.
• K=3,4: There is none.

Sample Input 2

6
1 1 2 2 5

Sample Output 2

6
6
2
0
0
0

Sample Input 3

6
1 1 1 1 1

Sample Output 3

6
10
10
5
1
0

Sample Input 4

10
1 2 1 2 1 1 2 6 9

Sample Output 4

10
30
47
38
16
3
0
0
0
0

见题面 Input 部分

见题面 Output 部分