think-cell Round 1 |
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Finished |
Stack has an array $$$a$$$ of length $$$n$$$ such that $$$a_i = i$$$ for all $$$i$$$ ($$$1 \leq i \leq n$$$). He will select a positive integer $$$k$$$ ($$$1 \leq k \leq \lfloor \frac{n-1}{2} \rfloor$$$) and do the following operation on $$$a$$$ any number (possibly $$$0$$$) of times.
Stack wonders how many arrays $$$a$$$ can he end up with for each $$$k$$$ ($$$1 \leq k \leq \lfloor \frac{n-1}{2} \rfloor$$$). As Stack is weak at counting problems, he needs your help.
Since the number of arrays might be too large, please print it modulo $$$998\,244\,353$$$.
$$$^\dagger$$$ A sequence $$$x$$$ is a subsequence of a sequence $$$y$$$ if $$$x$$$ can be obtained from $$$y$$$ by deleting several (possibly, zero or all) elements. For example, $$$[1, 3]$$$, $$$[1, 2, 3]$$$ and $$$[2, 3]$$$ are subsequences of $$$[1, 2, 3]$$$. On the other hand, $$$[3, 1]$$$ and $$$[2, 1, 3]$$$ are not subsequences of $$$[1, 2, 3]$$$.
Each test contains multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 2 \cdot 10^3$$$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$3 \leq n \leq 10^6$$$) — the length of the array $$$a$$$.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^6$$$.
For each test, on a new line, print $$$\lfloor \frac{n-1}{2} \rfloor$$$ space-separated integers — the $$$i$$$-th integer representing the number of arrays modulo $$$998\,244\,353$$$ that Stack can get if he selects $$$k=i$$$.
434510
2 4 10 2 487 162 85 10
In the first test case, two $$$a$$$ are possible for $$$k=1$$$:
In the second test case, four $$$a$$$ are possible for $$$k=1$$$:
In the third test case, two $$$a$$$ are possible for $$$k=2$$$:
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