> For the complete documentation index, see [llms.txt](https://docs.1l.lol/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.1l.lol/advanced-math/pascal-triangle.md).

# 杨辉三角

### 定义

每行两端为 1，其余数等于上一行相邻两数之和：

```
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
```

### 与组合数的关系

第 n 行第 k 个数正好是组合数 C(n-1, k-1)（从 0 开始编号）。

### 实现

```cpp
const int N = 30;
long long c[N][N];
for (int i = 0; i < N; i++) {
    c[i][0] = c[i][i] = 1;
    for (int j = 1; j < i; j++)
        c[i][j] = c[i-1][j-1] + c[i-1][j];
}
cout << c[n][k];   // 即 C(n, k)
```

### 应用

* 小范围组合数打表 O(n²)
* 二项式 (a+b)^n 的系数
* 路径计数问题


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