popcount <= 2

AtCoder

IDarc200_e

Time3000ms

Memory256MB

Difficulty—

You are given positive integers $N,M$. Find the number, modulo $998244353$, of non-negative integer sequences $A=(A_1,A_2,\ldots,A_N)$ of length $N$ that satisfy all the following conditions. * $0\le A_i < 2^M$ $(1\le i\le N)$ * $\operatorname{popcount}(A_i\oplus A_j) \le 2$ $(1\le i < j\le N)$ Here, for non-negative integers $x,y$, we denote by $x\oplus y$ the bitwise $\mathrm{XOR}$ operation of $x$ and $y$, and by $\operatorname{popcount}(x)$ the popcount of $x$. You are given $T$ test cases, so find the answer for each. What is bitwise $\mathrm{XOR}$ operation?The bitwise $\mathrm{XOR}$ of non-negative integers $A, B$, denoted $A \oplus B$, is defined as follows. * When $A \oplus B$ is written in binary, the digit in the $2^k$ ($k \geq 0$) place is $1$ if exactly one of $A, B$ has $1$ in the $2^k$ place when written in binary, and $0$ otherwise. For example, $3 \oplus 5 = 6$ (in binary: $011 \oplus 101 = 110$). What is popcount?For a non-negative integer $x$, $\operatorname{popcount}(x)$ is the number of $1$s when $x$ is written in binary. More precisely, for a non-negative integer $x$ such that $\displaystyle x=\sum _ {i=0} ^ \infty b _ i2 ^ i\ (b _ i\in\lbrace0,1\rbrace)$, we have $\displaystyle\operatorname{popcount}(x)=\sum _ {i=0} ^ \infty b _ i$. For example, $13$ in binary is `1101`, so we have $\operatorname{popcount}(13)=3$. ## Constraints * $1\le T\le 2\times 10^5$ * $2\le N,M < 998244353$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $T$ $\text{case}_1$ $\text{case}_2$ $\vdots$ $\text{case}_T$ Each test case is given in the following format: $N$ $M$ [samples]

Samples

Input #1

Output #1

56
16384
549499339

Consider the first test case.
For example, $A=(4,5)$ satisfies the conditions because each element is between $0$ and $2^3-1=7$, and $\operatorname{popcount}(4\oplus 5)=\operatorname{popcount}(1)=1\le 2$. On the other hand, neither $A=(2,5)$ nor $A=(0,7)$ satisfies the conditions.
There are $56$ sequences $A$ that satisfy the conditions, so output $56$ on the first line.

API Response (JSON)

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