201043: [AtCoder]ARC104 D - Multiset Mean

Problem Statement

Given positive integers $N, K$ and $M$, solve the following problem for every integer $x$ between $1$ and $N$ (inclusive):

• Find the number, modulo $M$, of non-empty multisets containing between $0$ and $K$ (inclusive) instances of each of the integers $1, 2, 3 \cdots, N$ such that the average of the elements is $x$.

Constraints

• $1 \leq N, K \leq 100$
• $10^8 \leq M \leq 10^9 + 9$
• $M$ is prime.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $K$ $M$

Output

Use the following format:

$c_1$
$c_2$
$:$
$c_N$

Here, $c_x$ should be the number, modulo $M$, of multisets such that the average of the elements is $x$.

Sample Input 1

3 1 998244353

Sample Output 1

1
3
1

Consider non-empty multisets containing between $0$ and $1$ instance(s) of each of the integers between $1$ and $3$. Among them, there are:

• one multiset such that the average of the elements is $k = 1$: $\{1\}$;
• three multisets such that the average of the elements is $k = 2$: $\{2\}, \{1, 3\}, \{1, 2, 3\}$;
• one multiset such that the average of the elements is $k = 3$: $\{3\}$.

Sample Input 2

1 2 1000000007

Sample Output 2

2

Consider non-empty multisets containing between $0$ and $2$ instances of each of the integers between $1$ and $1$. Among them, there are:

• two multisets such that the average of the elements is $k = 1$: $\{1\}, \{1, 1\}$.

Sample Input 3

10 8 861271909

Sample Output 3

8
602
81827
4054238
41331779
41331779
4054238
81827
602
8