Description
Return the adjacent factorial ratio or its reciprocal without constructing factorials.
A sequence rule specifies how terms relate. Checking one transition, advancing a recurrence, and bounding an infinite tail answer different questions. A calculation from supplied coefficients assumes that rule; it does not infer it from observations.
Return the adjacent factorial ratio or its reciprocal without constructing factorials.
When to use this tool
Use it for factorial sequence ratio at index when the required inputs and conventions are already available.
How to use the tool and how it works
- Enter index, backwards. Use the stated units and boundary conventions.
- Run the calculation. Correct any input error before interpreting a result.
- Read numerator, denominator. Inspect any known, defined, inclusion, or boundary flags before interpreting numeric values.
Represent the adjacent factorial ratio n!/(n-1)! as numerator n and denominator one, or its reciprocal when backwards is true. Index must be from one through 1,000,000.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| index | Zero-based selected index, unless a different convention is stated in the mechanics. | Required. Safe integer; minimum 1; maximum 1000000. |
| backwards | Backwards | Required. Boolean. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| numerator | Numerator | Required. Safe integer. |
| denominator | Denominator | Required. Safe integer. |
{
"input": {
"index": 5,
"backwards": false
},
"output": {
"numerator": 5,
"denominator": 1
}
}Limitations and assumptions
- No factorial is constructed and the ratio is not for two arbitrary factorial arguments. The caller chooses the ratio direction explicitly.
- The field tables describe structural bounds; the mechanics also state cross-field requirements. Malformed input shapes, disallowed keys, and invalid domains cause input errors. Results must remain finite and within the declared output contract. A successful false flag or zero placeholder is distinct from an error.
- Numeric arithmetic uses binary floating point. Exact comparisons have no hidden tolerance or decimal rounding; negative zero is exported as zero. Decimal-text integer operations retain the exactness specifically stated above.
Alternative or Complementary methods
For factorial sequence ratio at index, check the source records and the supplied rule before comparing the result with another method. Use exact integer or symbolic arithmetic when a numerical equality is insufficient.
References
Similar or alternative tools
- Factorial Ratio Adjacent Logterm
Calculate the natural logarithm of the ratio of consecutive factorials without constructing them.
- Figurate Sequence Inverse Index Discriminant
Calculate a polygonal-number discriminant and numerical inverse index, with a floating-point integrality check.