Score : $600$ points

问题描述

给定整数 $R$，$G$，$B$ 和 $K$。有多少个由 R，G 和 B 组成的字符串 $S$ 满足以下所有条件？请计算满足条件的字符串数量模 $998244353$ 的结果。

• 字符串 $S$ 中 R、G 和 B 出现的次数分别为 $R$，$G$ 和 $B$。
• 字符串 $S$ 中作为（连续）子串出现的 RG 的次数为 $K$。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given integers $R$, $G$, $B$, and $K$. How many strings $S$ consisting of R, G, and B satisfy all of the conditions below? Find the count modulo $998244353$.

• The number of occurrences of R, G, and B in $S$ are $R$, $G$, and $B$, respectively.
• The number of occurrences of RG as (contiguous) substrings in $S$ is $K$.

Constraints

• $1 \leq R,G,B\leq 10^6$
• $0 \leq K \leq \mathrm{min}(R,G)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$R$ $G$ $B$ $K$

Output

Print the answer.

Sample Input 1

2 1 1 1

Sample Output 1

6

The following six strings satisfy the conditions.

• RRGB
• RGRB
• RGBR
• RBRG
• BRRG
• BRGR

Sample Input 2

1000000 1000000 1000000 1000000

Sample Output 2

80957240

Find the count modulo $998244353$.

update @ 2024/3/10 11:15:28