Catalan Number Calculator

A Catalan number counts several different combinatorial structures that all happen to share exactly the same count, which is part of what makes the sequence worth knowing. The number of structurally distinct full binary trees built from n+1 leaves, the number of monotonic lattice paths from one corner of an n-by-n grid to the opposite corner that never rise above the diagonal, and the number of ways to fully and validly parenthesize a string of n+1 factors are all the identical number, C(n). This calculator gets there through the closed-form formula C(n) equals C(2n, n) divided by (n + 1), computing the central binomial coefficient first and then dividing, rather than actually building and counting trees or paths. Raising n does not just raise the count — it raises it at an accelerating rate, since the ratio between one Catalan number and the next climbs steadily toward 4 as n grows (it is 2.5 at n = 3, reaches exactly 3.5 at n = 11, and hits roughly 3.81 by n = 30) without ever quite touching 4.

The input tops out at n = 30 for a concrete reason: that is the largest index whose true Catalan number is still exactly representable as a JavaScript number. One step further, at n = 31, the true value already exceeds the largest integer double-precision floating point can represent without rounding error, so the field simply will not accept a larger index. Missing from this calculator: it never lists the actual binary trees, lattice paths, or parenthesizations it is counting — only the count itself — and there is no way to jump straight to one very large index without rendering the whole climbing sequence up to it in the chart.