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Contest: Task: Related: TaskG TaskI

Score : $600$ points

Problem Statement

There are $N$ balls numbered $1$ through $N$. Ball $i$ is painted in Color $a_i$.

For a permutation $P = (P_1, P_2, \dots, P_N)$ of $(1, 2, \dots, N)$, let us define $C(P)$ as follows:

  • The number of pairs of adjacent balls with different colors when Balls $P_1, P_2, \dots, P_N$ are lined up in a row in this order.

Let $S_N$ be the set of all permutations of $(1, 2, \dots, N)$. Also, let us define $F(k)$ by:

$\displaystyle F(k) = \left(\sum_{P \in S_N}C(P)^k \right) \bmod 998244353$.

Enumerate $F(1), F(2), \dots, F(M)$.

Constraints

  • $2 \leq N \leq 2.5 \times 10^5$
  • $1 \leq M \leq 2.5 \times 10^5$
  • $1 \leq a_i \leq N$
  • All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$
$a_1$ $a_2$ $\dots$ $a_N$

Output

Print the answers in the following format:

$F(1)$ $F(2)$ $\dots$ $F(M)$

Sample Input 1

3 4
1 1 2

Sample Output 1

8 12 20 36

Here is the list of all possible pairs of $(P, C(P))$.

  • If $P=(1,2,3)$, then $C(P) = 1$.
  • If $P=(1,3,2)$, then $C(P) = 2$.
  • If $P=(2,1,3)$, then $C(P) = 1$.
  • If $P=(2,3,1)$, then $C(P) = 2$.
  • If $P=(3,1,2)$, then $C(P) = 1$.
  • If $P=(3,2,1)$, then $C(P) = 1$.

We may obtain the answers by assigning these values into $F(k)$. For instance, $F(1) = 1^1 + 2^1 + 1^1 + 2^1 + 1^1 + 1^1 = 8$.

Sample Input 2

2 1
1 1

Sample Output 2

0

Sample Input 3

10 5
3 1 4 1 5 9 2 6 5 3

Sample Output 3

30481920 257886720 199419134 838462446 196874334