Exact Integer Catalan Number Calculator

Start at C0 = 1 and apply the exact multiplicative recurrence C(n+1) = C(n) 2(2n+1)/(n+2). Zero indexing matters: C0 and C1 are both one, so confirm the intended convention before comparing tables. Inputs must be whole numbers inside the displayed safe bound. The calculator does not round decimals into the domain or silently substitute a different convention, so verify each entered integer before interpreting the exact result.

Catalan numbers count specific noncrossing or balanced families; they do not count every tree, path, or parenthesization variant without the stated constraints. The binomial expression and recurrence provide two independent ways to verify a modest term. Use the displayed factors, recurrence step, or modular residue as an independent check. Exact integer output means the implemented arithmetic has no decimal approximation within its bound; it does not prove an unrelated theorem or an unstated combinatorial model.

The calculator stops at n = 25 before larger Catalan values exceed reliable exact integer representation in this workflow. Arbitrary-precision arithmetic is required beyond this page's bound if an exact integer, rather than a magnitude estimate, is needed. The bound keeps browser work predictable and every returned integer within JavaScript's exact safe range. Larger inputs need arbitrary-precision software, explicit resource controls, and independent verification rather than treating an overflowed floating-point value as exact.