Pascal Triangle Row
Return the zero-indexed nth row of Pascal’s triangle as exact decimal binomial coefficients; n is capped at 1,000 to bound the output.
Description
Pascal’s triangle lists binomial coefficients by row.
Pascal’s triangle lists binomial coefficients by row. Row n contains n+1 coefficients, including one at each end.1
The tool returns coefficients. The indexing and output conventions below determine how to read these values.
When to use Pascal Triangle Row
Use this for the following task: Return the zero-indexed nth row of Pascal’s triangle as exact decimal binomial coefficients; n is capped at 1,000 to bound the output. Use the input and output conventions below to reproduce the intended mathematical quantity.
How to use Pascal Triangle Row and understand its results
- Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
- Enter n according to the field table, then run the tool.
- Read coefficients. Preserve exact decimal strings and the stated indexing or coefficient order.
- Keep exact decimal strings for subsequent integer arithmetic; distinguish them from any explicitly approximate numeric result.
| Input field | Meaning | Value type and bounds | Requirement |
|---|---|---|---|
| n | Zero-indexed row (resource limit 1000) | integer; minimum 0; maximum 1000 | Required |
Start with BigInt one, append its decimal text and update by the exact product-and-division recurrence for each subsequent coefficient.
row[n][k]=C(n,k); nextCoefficient=current×(n−k)/(k+1) The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| coefficients | Exact row coefficients | list of string (integer/arbitrary) |
{
"input": {
"n": 5
},
"output": {
"coefficients": [
"1",
"5",
"10",
"10",
"5",
"1"
]
}
}Limitations and assumptions
The row is zero-based and at most 1000. Exact values are strings, so downstream arithmetic must parse them rather than concatenate them.
Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.
Alternative or Complementary methods
Choose Binomial Coefficient Modulo a Prime by Lucas’s Theorem when you need its operation: Compute C(n,k) modulo a prime from the base-prime digits of exact nonnegative decimal integers; the prime is capped at 10,000 for bounded work. Choose Multinomial Coefficient for the complementary operation: Count arrangements with the supplied nonnegative category counts using exact arithmetic; at most 128 categories and 2,000 total elements are admitted.
Background reference: Digital Library of Mathematical Functions, §26.3.
References
-
Digital Library of Mathematical Functions, §26.3 — National Institute of Standards and Technology
Similar or alternative tools
- Binomial Coefficient Modulo a Prime by Lucas’s Theorem
Compute C(n,k) modulo a prime from the base-prime digits of exact nonnegative decimal integers; the prime is capped at 10,000 for bounded work.
- Multinomial Coefficient
Count arrangements with the supplied nonnegative category counts using exact arithmetic; at most 128 categories and 2,000 total elements are admitted.