🔺 杨辉三角（帕斯卡三角形）生成器Pascal's Triangle Generator
用大整数精确生成杨辉三角前 1–100 行（居中三角、左对齐或 CSV），或单独算出第 0–1000 行的全部二项式系数；还能显示奇偶图案，看到其中的谢尔宾斯基三角形。Generate the first 1–100 rows of Pascal's triangle exactly with big integers — centred, left-aligned or as CSV — or get every binomial coefficient in a single row from 0 to 1000. An odd/even view reveals the Sierpiński triangle inside it.
使用说明与常见问题How-to & FAQ
「杨辉三角（帕斯卡三角形）生成器」是什么？What is the Pascal's Triangle Generator tool?
用大整数精确生成杨辉三角前 1–100 行（居中三角、左对齐或 CSV），或单独算出第 0–1000 行的全部二项式系数；还能显示奇偶图案，看到其中的谢尔宾斯基三角形。Generate the first 1–100 rows of Pascal's triangle exactly with big integers — centred, left-aligned or as CSV — or get every binomial coefficient in a single row from 0 to 1000. An odd/even view reveals the Sierpiński triangle inside it.
如何使用「杨辉三角（帕斯卡三角形）生成器」？How do I use the Pascal's Triangle Generator tool?
在「杨辉三角（帕斯卡三角形）生成器」中，填写「模式」、「行数 n（1–100）」、「格式」、「行号 r（0–1000）」等参数，结果会随输入实时计算并显示在下方。In the Pascal's Triangle Generator: Fill in "Mode", "Rows, n (1–100)", "Format", "Row number, r (0–1000)"; results are recalculated instantly as you type.
杨辉三角是怎么构成的？How is Pascal's triangle built?
每行首尾都是 1，中间每个数等于它左上和右上两个数之和。第 r 行（从第 0 行算起）第 k 个数就是二项式系数 C(r, k)，也就是 (a + b)ʳ 展开后各项的系数，所以第 r 行所有数之和是 2ʳ。Each row starts and ends with 1, and every other number is the sum of the two above it. Entry k of row r (counting from row 0) is the binomial coefficient C(r, k) — the coefficients of (a + b)ʳ — so row r adds up to 2ʳ.
为什么叫杨辉三角？Why is it also called Yang Hui's triangle?
南宋数学家杨辉在 1261 年的《详解九章算法》中记载了这个三角形，并注明出自北宋贾宪，比法国数学家帕斯卡（1654 年）早约 400 年，所以在中国称为杨辉三角或贾宪三角。The Song-dynasty mathematician Yang Hui recorded it in 1261, crediting the earlier Jia Xian — about 400 years before Blaise Pascal's 1654 treatise — so in China it is called Yang Hui's (or Jia Xian's) triangle.
奇偶图案是什么？What is the odd/even pattern?
把奇数画成 *、偶数留空，杨辉三角会呈现谢尔宾斯基三角形这种分形：行数取 2 的幂（如 16、32、64）时图案最完整。Drawing odd entries as * and leaving even ones blank turns the triangle into the Sierpiński fractal; it looks most complete when the number of rows is a power of 2 (16, 32, 64).
大数会不会算错？Are the big numbers exact?
会精确到每一位。每个数都用 JavaScript 大整数（BigInt）按 C(r, k) = C(r, k−1) × (r − k + 1) / k 逐个计算，例如第 200 行中间的 C(200, 100) 有 59 位，结果和 Python 的 math.comb 一致。Yes, to the last digit. Each entry is computed with JavaScript big integers (BigInt) using C(r, k) = C(r, k−1) × (r − k + 1) / k; the middle of row 200, C(200, 100), has 59 digits and matches Python's math.comb.
「杨辉三角（帕斯卡三角形）生成器」免费吗？需要注册吗？Is the Pascal's Triangle Generator free? Do I need an account?
「杨辉三角（帕斯卡三角形）生成器」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Pascal's Triangle Generator is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.