Pascal's triangle generator
Ask for 5 rows and get Pascal's triangle down to 1, 4, 6, 4, 1: fifteen exact numbers in which each entry is the sum of the two above it. Ready for worksheets and binomial expansions. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- N
- 5
- Row count
- 5
- Entry count
- 15
Pascal's triangle is the classical arrangement of binomial coefficients used in algebra, probability, combinatorics, and discrete mathematics. Each interior number is the sum of the two numbers above it, so row r encodes the coefficients of (a+b) raised to the power r minus one. Students meet it when expanding polynomials, counting paths on a grid, or reading the coefficients that appear in the binomial theorem. Requests with n below one fail immediately with a clear validation error.
What Pascal's triangle is and when you need the first n rows
Pascal's triangle organizes binomial coefficients so that the entry in row r at zero-based index k equals C of r minus one choose k. The top row is a single one. Every later row begins and ends with one, and each interior entry is the sum of the two parents sitting above-left and above-right in the previous row.
That recurrence is identical to the Pascal identity C of m choose k equals C of m minus one choose k minus one plus C of m minus one choose k, which is why the triangle appears whenever you expand (a plus b) to the m, count subsets of size k from m items, or walk lattice paths that only move right and up.
Classroom worksheets often ask for the first five, ten, or twenty rows so learners can spot symmetry, the powers of two along the row sums, and the hockey-stick identity in the diagonals. Probability courses use the same rows as coefficients of a fair coin expansion. Programming interviews and contest problems sometimes require printing the triangle under a size bound without floating-point error on the center entries.
A generator that returns the full block of rows, not a single coefficient, matches those search intents: people look for a Pascal triangle generator, binomial coefficient table, or first n rows calculator, not only an isolated C of n choose k function.
What you can do with it
Teaching the binomial theorem and Pascal identity
Lesson pages can show exact triangle rows beside the identity C(m,k)=C(m−1,k−1)+C(m−1,k) so learners connect the recurrence to polynomial expansions.
FAQ
What is the relationship between Pascal's triangle and binomial coefficients?
The entry in row r (1-based) at zero-based index k is C(r−1, k). Interior entries follow the Pascal identity: each is the sum of the two parents above it.
What happens if n is below 1?
Pascal's triangle generation requires a positive row count; those calls are not charged.
Is the Pascal's triangle generator free?
The browser tool runs free on your device.
How many rows can I generate at once?
From 1 up to 500 rows. Every number stays exact, even the large ones in the middle of the lower rows, so you can print or copy the block without rounding.
What do the rows of Pascal's triangle add up to?
Each row adds up to a power of two, so every row total is double the one before. Spotting this pattern, along with the symmetry of each row, is a classic classroom exercise.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/combo/pascal-triangle \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"n":5}'const res = await fetch("https://api.kit.forhosting.com/combo/pascal-triangle", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"n": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/combo/pascal-triangle",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"n": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/combo/pascal-triangle", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"n":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"n":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/combo/pascal-triangle", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"n": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "combo.pascal_triangle",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Limits
max_n | 500 |
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
This capability returns the first n rows as structured JSON with every entry computed in BigInt so large middle coefficients never lose digits.
What Pascal's triangle is and when you need the first n rows
Delivering exact integers as JSON makes the result easy to paste into notes, diff in tests, or render in a custom layout without reimplementing the recurrence in every language you teach.
How this generator builds rows, validates n, and returns exact entries
The engine checks that n is present as a finite integer at least one and at most the published upper bound of five hundred. For accepted n the implementation allocates n rows and fills each row using BigInt arithmetic: the first and last positions are one, and each interior position is the sum of the two neighboring entries from the previous row. That approach never multiplies large factorials and never divides, so every coefficient stays exact even when middle entries exceed ordinary double precision. Edge cases are explicit: n equals one yields a single row containing one; n equals two yields one and then one one. Send a JSON body with the required positive integer field n. Values below one fail immediately with invalid_input because a non-positive row count is outside the published domain for this capacity, and those failed calls are not charged. Missing fields, non-integers, and values above the limit also fail with invalid_input. Each coefficient is returned as a JSON number when it fits inside the safe integer range, otherwise as a decimal string so clients never lose high-order digits. The response echoes n, includes the full rows array, and adds row_count plus entry_count equal to n times n plus one over two for convenience.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate exact table of the first n rows while studying the binomial theorem, checking a contest answer, or preparing a handout without installing a computer-algebra system. Accepted row counts run from one through five hundred so classroom demos and combinatorial tables remain feasible without unbounded memory growth on pathological sizes. There is no language model, no external arithmetic service, and no randomness, so the capability is safe to pin in continuous-integration checks and reproducible lesson materials. Each successful request is one billable item at $0.002; failed validation does not consume balance. Results are fully deterministic: the same n always yields the same rows, row_count, and entry_count fields. Wire the endpoint into combinatorics demos, coding-interview sandboxes, probability tutorials that need binomial expansions, or integer preprocessing steps where you already prefer exact Pascal arithmetic over approximate heuristics or opaque black-box widgets that cannot show the row recurrence or the domain restriction when n is below one.
Integration use cases
Homework and contest verification
Students and graders confirm the first n rows of binomial coefficients without a local CAS, including edge cases n=1 and larger tables where center entries exceed safe integers.
Deterministic fixtures for tests and demos
Pipelines that seed unit tests or interactive combinatorics demos can call a pure Pascal oracle instead of reimplementing BigInt row recurrence in every language.
Technical questions
How do I generate Pascal's triangle with this API?
Send a JSON body with integer field n (number of rows) to combo.pascal_triangle. The response returns rows as nested arrays of exact coefficients, plus row_count and entry_count.
Why are some entries returned as strings?
Coefficients that exceed Number.MAX_SAFE_INTEGER are returned as decimal strings so JSON clients keep full exact precision via BigInt. Smaller values stay JSON numbers.