Catalan Number CalculatorFree online math calculator — instant results, no sign-up
Calculate a Catalan number using exact integer arithmetic.
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About the Catalan Number Calculator
The Catalan Number Calculator computes the Catalan number C_n for a nonnegative integer index using exact integer arithmetic. Catalan numbers count structures such as valid arrangements of matched parentheses and full binary tree shapes. With the default index of 10, this calculator returns 16796.
- Index
- The index n is a nonnegative whole number identifying the Catalan number to calculate; it has no units and defaults to 10.
Formula and method
C_n = binom(2n, n) / (n + 1) = (2n)! / [n! × n! × (n + 1)]. For n = 10: C_10 = binom(20, 10) / 11 = 184756 / 11 = 16796.
- nThe nonnegative integer index entered by the user.
- C_nThe Catalan number at index n, a dimensionless integer count.
- binom(2n, n)The binomial coefficient counting ways to choose n items from 2n items.
- !The factorial operator: the product of the positive integers up to its argument, with 0! defined as 1.
Validate that the index is a nonnegative integer, compute the central binomial coefficient binom(2n, n) with exact integer arithmetic, and divide it exactly by n + 1. The result is always an integer for a valid index, so no rounding or floating-point approximation is needed.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Index10
Results
- Catalan number16796
How to use the Catalan Number Calculator
- 1Enter a nonnegative whole number in Index, or leave the default value of 10.
- 2Check that the index has no fractional part and is not negative.
- 3Run the calculation to evaluate C_n = binom(2n, n) / (n + 1).
- 4Read the Catalan number output; for Index = 10, it is 16796.
Reading your result
The output row labeled Catalan number shows the exact value of C_n for the entered index. At Index = 10, the output 16796 means there are 16796 valid arrangements of 10 pairs of matched parentheses, or equivalently 16796 full ordered binary tree shapes with 10 internal nodes. The output is a count, not a measurement or percentage.
Limitations
The index must be a nonnegative integer; negative and fractional indices are outside this calculator's scope. Indexing begins at C_0 = 1. Exact integer arithmetic avoids rounding error, but Catalan numbers grow rapidly, so very large indices may require substantial processing time, memory, or output space. No specific maximum supported index is stated here. Catalan counts apply only to structures satisfying the relevant ordering and matching rules, not to every kind of tree or arrangement.
Common questions
Why does Index = 10 give 16796?+
The central binomial coefficient binom(20, 10) is 184756. Dividing it by 10 + 1 gives the exact Catalan number 16796.
Does the Catalan sequence start at index 0 or index 1?+
This calculator uses zero-based indexing, with C_0 = 1. That value represents one empty structure, such as the empty arrangement of matched parentheses.
Why is division by n + 1 guaranteed to produce an integer?+
For a nonnegative integer n, binom(2n, n) / (n + 1) equals binom(2n, n) − binom(2n, n + 1). Since both binomial coefficients are integers, their difference is an integer.
Does C_10 count every possible ordering of 10 opening and 10 closing parentheses?+
No. It counts only properly matched arrangements: every opening parenthesis must be closed, and no prefix may contain more closing than opening parentheses. Exactly 16796 arrangements satisfy those rules.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.