Description
Return adjacent binomial ratio and its natural logarithm.
Exact combinatorial recurrences distinguish individual summands, adjacent ratios and complete totals. A local arithmetic identity can be correct without proving that its supplied inputs are the intended counts or that a whole enumeration has been completed.
Return adjacent binomial ratio and its natural logarithm.
When to use this tool
Use Binomial Coefficient Log Ratio Step when checking the stated recurrence, coefficient-ratio or inclusion-exclusion term.
How to use the tool and how it works
- Enter n, k. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read ratio, logRatio. Use the output convention below.
Return adjacent binomial ratio and its natural logarithm. Require n 1..1000000 and k in [0,n); ratio = (n-k)/(k+1), logRatio = ln(ratio).1
Pascal recurrence and adjacent-coefficient ratios are exact combinatorial identities; checking a supplied local identity does not verify a complete row.2
| Input field | Meaning and units | Accepted input |
|---|---|---|
| n | N | Required. Safe integer; minimum 1; maximum 1000000. |
| k | K | Required. Safe integer; minimum 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| ratio | Ratio | Required. Finite number. |
| logRatio | Log Ratio | Required. Finite number. |
{
"input": {
"n": 5,
"k": 2
},
"output": {
"ratio": 1,
"logRatio": 0
}
}Limitations and assumptions
- Direction is k+1 over k for fixed n; this does not compute complete coefficients or a log-probability mass function.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Use the complete recurrence with justified initial conditions and exact integer arithmetic when a full counting result is required.
References
Similar or alternative tools
- Binomial Lattice Step Forward Ratio
Return the forward ratio of adjacent binomial coefficients.
- Combinatorics: Binomial Adjacent Coefficient Ratio
Return one adjacent binomial coefficient ratio.