Description
Return one exact Catalan-convolution summand.
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 one exact Catalan-convolution summand.
When to use this tool
Use Catalan Convolution Single Summand when checking the stated recurrence, coefficient-ratio or inclusion-exclusion term.
How to use the tool and how it works
- Enter order, index. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read term. Use the output convention below.
Return one exact Catalan-convolution summand. Require whole order 1..64 and index in [0,order); compute C_i*C_(order-1-i) with C_n = binomial(2n,n)/(n+1), returning decimal term text.1
The exact binomial coefficient uses the shared calculation behind Pascal row; this helper returns only the local factor or summand described above.2
Catalan numbers have a central-binomial formula and a convolution recurrence; this result is one summand of that recurrence.3
| Input field | Meaning and units | Accepted input |
|---|---|---|
| order | Order | Required. Safe integer; minimum 1; maximum 64. |
| index | Position under the explicit indexing convention; inspect any missing-value sentinel. | Required. Safe integer; minimum 0. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| term | Exact C_index*C_(order-1-index) as decimal integer text, not the complete convolution sum. | Required. Text; maximum UTF-16 length 8192. |
{
"input": {
"order": 4,
"index": 1
},
"output": {
"term": "2"
}
}Limitations and assumptions
- This is one nonnegative summand, not the full Catalan number at order; no triangulation or tree is generated.
- 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 Pascal Single Recurrence Residual
Check an exact Pascal recurrence residual.
- Binomial Lattice Step Forward Ratio
Return the forward ratio of adjacent binomial coefficients.