Counting 101
Description
It's been a long summer's day, with the constant chirping of cicadas and the heat which never seemed to end. Finally, it has drawn to a close. The showdown has passed, the gates are open, and only a gentle breeze is left behind.
Your predecessors had taken their final bow; it's your turn to take the stage.
Sorting through some notes that were left behind, you found a curious statement named Problem 101:
- Given a positive integer sequence $a_1,a_2,\ldots,a_n$, you can operate on it any number of times. In an operation, you choose three consecutive elements $a_i,a_{i+1},a_{i+2}$, and merge them into one element $\max(a_i+1,a_{i+1},a_{i+2}+1)$. Please calculate the maximum number of operations you can do without creating an element greater than $m$.
After some thought, you decided to propose the following problem, named Counting 101:
- Given $n$ and $m$. For each $k=0,1,\ldots,\left\lfloor\frac{n-1}{2}\right\rfloor$, please find the number of integer sequences $a_1,a_2,\ldots,a_n$ with elements in $[1, m]$, such that when used as input for Problem 101, the answer is $k$. As the answer can be very large, please print it modulo $10^9+7$.
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1\le t\le10^3$). The description of the test cases follows.
The only line of each test case contains two integers $n$, $m$ ($1\le n\le 130$, $1\le m\le 30$).
For each test case, output $\left\lfloor\frac{n+1}{2}\right\rfloor$ numbers. The $i$-th number is the number of valid sequences such that when used as input for Problem 101, the answer is $i-1$, modulo $10^9+7$.
Input
Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1\le t\le10^3$). The description of the test cases follows.
The only line of each test case contains two integers $n$, $m$ ($1\le n\le 130$, $1\le m\le 30$).
Output
For each test case, output $\left\lfloor\frac{n+1}{2}\right\rfloor$ numbers. The $i$-th number is the number of valid sequences such that when used as input for Problem 101, the answer is $i-1$, modulo $10^9+7$.
2
3 2
10 10
6 2
1590121 23399118 382293180 213020758 379696760
Note
In the first test case, there are $2^3=8$ candidate sequences. Among them, you can operate on $[1,2,1]$ and $[1,1,1]$ once; you cannot operate on the other $6$ sequences.