Calculator Inputs
Large screens show three columns, medium screens show two, and mobile shows one.
Detailed Results
| Metric | Symbol | Computed Value | Explanation |
|---|---|---|---|
| Target Index | n | 10 | The selected term position in the Catalan sequence. |
| Catalan Number | Cn | 16796 | The number of valid structures counted by the nth Catalan term. |
| Previous Term | Cn-1 | 4862 | The immediate preceding Catalan value for growth comparison. |
| Next Term | Cn+1 | 58786 | The following sequence term obtained from the closed-form definition. |
| Central Binomial Coefficient | (2n choose n) | 184756 | The combinatorial core used before division by n + 1. |
| Growth Ratio | Cn/Cn-1 | 3.454545 | Shows how rapidly the sequence grows between adjacent indices. |
| Asymptotic Estimate | 4n / (n3/2√π) | 18707.897292 | A large-n approximation commonly used in analysis. |
| Approximation Error | % error | 11.3831% | Relative difference between the estimate and exact Catalan value. |
| Comparison Term | Ck | 5 when k = 3 | A custom comparison term for sequence exploration. |
| Cumulative Range Sum | ΣCi | 13402697 | Total of all Catalan numbers between the selected bounds. |
Plotly Graph
The graph displays Catalan values across the chosen range for quick visual growth analysis.
Example Data Table
| n | Catalan Number | Example Interpretation |
|---|---|---|
| 0 | 1 | One empty valid structure. |
| 1 | 1 | One binary tree with one internal node. |
| 2 | 2 | Two balanced parenthesis arrangements. |
| 3 | 5 | Five full binary tree shapes. |
| 4 | 14 | Fourteen triangulations of a pentagon. |
| 5 | 42 | Forty-two noncrossing partition patterns. |
| 6 | 132 | One hundred thirty-two valid recursive structures. |
Formula Used
Closed form: Cn = (1 / (n + 1)) × (2n choose n)
Equivalent factorial form: Cn = (2n)! / ((n + 1)! × n!)
Recurrence: C0 = 1 and Cn+1 = [2(2n + 1) / (n + 2)] × Cn
Asymptotic estimate: Cn ≈ 4n / (n3/2√π)
Catalan numbers count many recursive combinatorial objects, including valid parentheses, binary tree shapes, monotonic lattice paths below a diagonal, and polygon triangulations. This calculator computes the exact nth value using the closed form and also compares it with a large-index approximation.
How to Use This Calculator
- Enter the target index n for the Catalan number you want.
- Set the starting and ending bounds for the plotted sequence range.
- Enter a comparison index k to inspect another Catalan term.
- Click Calculate Catalan Values to show the result above the form.
- Review the metrics table, graph, and sequence range output.
- Use Download CSV for spreadsheet export.
- Use Download PDF to save a clean report.
Generated Sequence Range
| Index n | Catalan Number | Running Sum |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 1 | 2 |
| 2 | 2 | 4 |
| 3 | 5 | 9 |
| 4 | 14 | 23 |
| 5 | 42 | 65 |
| 6 | 132 | 197 |
| 7 | 429 | 626 |
| 8 | 1430 | 2056 |
| 9 | 4862 | 6918 |
| 10 | 16796 | 23714 |
| 11 | 58786 | 82500 |
| 12 | 208012 | 290512 |
| 13 | 742900 | 1033412 |
| 14 | 2674440 | 3707852 |
| 15 | 9694845 | 13402697 |
FAQs
1. What does the Catalan sequence count?
Catalan numbers count many recursive structures, such as balanced parentheses, full binary trees, noncrossing partitions, polygon triangulations, and monotone paths that stay below a diagonal.
2. Why does the calculator use a closed-form formula?
The closed form gives exact values efficiently for moderate indices. It avoids slower recursive expansion and is ideal for showing combinations, ratios, and comparison metrics.
3. What is the recurrence relation used here?
The recurrence is C₀ = 1 and Cₙ₊₁ = 2(2n + 1)Cₙ / (n + 2). It lets you build the sequence term by term from earlier results.
4. Why does the sequence grow so quickly?
Catalan numbers combine factorial-like growth with division by n + 1. The sequence still rises rapidly, especially after moderate indices, which is visible in the graph.
5. What does the asymptotic estimate mean?
It is a large-index approximation. It becomes more useful as n increases and helps compare exact sequence values with analytical growth behavior.
6. Can I use this calculator for programming problems?
Yes. It is helpful for algorithm design, dynamic programming, parsing, binary tree analysis, lattice path counting, and interview problems involving recursive structures.
7. Why is there a comparison index k?
The extra index lets you compare another Catalan term without changing the main target. It is useful for observing differences in sequence growth across positions.
8. Are very large indices always exact?
This page uses floating-point computation, so very large values may lose integer precision. For moderate indices, results remain practical and highly informative.