Farmer John is interested in bracket sequences with common pairings.
A balanced bracket sequence is one in which every opening bracket is matched with a later closing bracket, and no prefix contains more closing brackets than opening brackets.
For a balanced bracket sequence $$$s$$$, a pairing is a pair of positions $$$(i,j)$$$ with $$$i \\lt j$$$ such that $$$s_i$$$ is an opening bracket, $$$s_j$$$ is a closing bracket, and the substring strictly between them is balanced.
Consider an ordered pair $$$(s,t)$$$ of balanced bracket sequences, where both $$$s$$$ and $$$t$$$ have length $$$2n$$$. A pairing $$$(i,j)$$$ is common to $$$s$$$ and $$$t$$$ if $$$(i,j)$$$ is a pairing in both sequences. For example, the bracket sequences $$$\\texttt{(()())}$$$ and $$$\\texttt{((()))}$$$ have one common pairing, namely the outer pair $$$(1,6)$$$.
Two ordered pairs $$$(s,t)$$$ and $$$(s',t')$$$ are considered different if $$$s \\ne s'$$$ or $$$t \\ne t'$$$.
For each $$$0 \\le k \\le n$$$, Farmer John wants to know the number of ordered pairs $$$(s,t)$$$ of balanced bracket sequences of length $$$2n$$$ with exactly $$$k$$$ common pairings. Since these numbers may be large, output them modulo $$$M$$$.
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \\le t \\le 500$$$). The description of the test cases follows.
The first line of each testcase will contain two integers, $$$n$$$ and $$$M$$$ ($$$1 \\leq n \\leq 500$$$, $$$10^8 \\leq M \\leq 10^9$$$) — half the length of the bracket sequence and the modulo to output the answer in. It is guaranteed that $$$M$$$ is prime.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$500$$$.
For each testcase, output $$$n + 1$$$ integers, the answer modulo $$$M$$$ for each $$$0 \\leq k \\leq n$$$.
51 9982443532 9982443533 9982443534 9982443538 998244353
0 12 0 28 12 0 562 64 56 0 14378790 646560 542960 296800 127400 34944 16016 0 1430
For the first test case, there is only one balanced bracket sequence with $$$n=1$$$: $$$\\texttt{()}$$$. It has exactly one pairing, namely $$$(1,2)$$$. Therefore, the only ordered pair of bracket sequences has exactly one common pairing, so the answer is $$$[0,1]$$$.
For the second test case, there are exactly two balanced bracket sequences with $$$n=2$$$: $$$a=\\texttt{(())}$$$ and $$$b=\\texttt{()()}$$$. The pairings of $$$a$$$ are $$$(1,4)$$$ and $$$(2,3)$$$, while the pairings of $$$b$$$ are $$$(1,2)$$$ and $$$(3,4)$$$.
Thus, the ordered pairs $$$(a,b)$$$ and $$$(b,a)$$$ have $$$0$$$ common pairings, while the ordered pairs $$$(a,a)$$$ and $$$(b,b)$$$ have $$$2$$$ common pairings. Therefore, the answer is $$$[2,0,2]$$$.