Score : $600$ points

问题陈述

在数轴上有三个人：第1个人、第2个人和第3个人。初始时刻（时间$t=0$），第1个人位于点$A$，第2个人位于点$B$，第3个人位于点$C$。其中，$A$、$B$和$C$均为整数，并且满足$A \equiv B \equiv C \pmod{2}$。

在时间$t=0$时，这三个人开始随机漫步。具体来说，在某一时刻$t$（$t$为非负整数）位于点$x$的人，在时刻$t+1$时以相同的概率移动到点$(x-1)$或点$(x+1)$。（所有移动的选择都是随机且相互独立的。）

求在时间$T$时，三人首次同时到达同一位置的概率对$998244353$取模的结果。

什么是模$998244353$下的有理数？我们可以证明所求概率始终是一个有理数。而且，在本题的约束条件下，当该值表示为两个互质整数$P$和$Q$的比值$\frac{P}{Q}$时，可以证明存在一个唯一的整数$R$，使得$R \times Q \equiv P\pmod{998244353}$且$0 \leq R < 998244353$。请找出这样的$R$。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

On a number line are person $1$, person $2$, and person $3$. At time $0$, person $1$ is at point $A$, person $2$ is at point $B$, and person $3$ is at point $C$.
Here, $A$, $B$, and $C$ are all integers, and $A \equiv B \equiv C \pmod{2}$.

At time $0$, the three people start random walks. Specifically, a person that is at point $x$ at time $t$ ($t$ is a non-negative integer) moves to point $(x-1)$ or point $(x+1)$ at time $(t+1)$ with equal probability. (All choices of moves are random and independent.)

Find the probability, modulo $998244353$, that it is at time $T$ that the three people are at the same point for the first time.

What is rational number modulo $998244353$? We can prove that the sought probability is always a rational number. Moreover, under the Constraints of this problem, when the value is represented as $\frac{P}{Q}$ by two coprime integers $P$ and $Q$, we can prove that there is a unique integer $R$ such that $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R \lt 998244353$. Find such $R$.

Constraints

• $0 \leq A, B, C, T \leq 10^5$
• $A \equiv B \equiv C \pmod{2}$
• $A, B, C$, and $T$ are integers.

Input

The input is given from Standard Input in the following format:

$A$ $B$ $C$ $T$

Output

Find the probability, modulo $998244353$, that it is at time $T$ that the three people are at the same point for the first time, and print the answer.

Sample Input 1

1 1 3 1

Sample Output 1

873463809

The three people are at the same point for the first time at time $1$ with the probability $\frac{1}{8}$. Since $873463809 \times 8 \equiv 1 \pmod{998244353}$, $873463809$ should be printed.

Sample Input 2

0 0 0 0

Sample Output 2

1

The three people may already be at the same point at time $0$.

Sample Input 3

0 2 8 9

Sample Output 3

744570476

Sample Input 4

47717 21993 74147 76720

Sample Output 4

844927176

update @ 2024/3/10 12:06:03