Exact Narayana Number Calculator

Evaluate N(n,k) = C(n,k)C(n,k-1)/n for one-based peak counts in Dyck paths of semilength n. Peak counts use the range one through n and partition the Catalan family. Inputs must be whole numbers inside the displayed domain; the page never rounds a decimal into an accepted index. This ties the answer to one explicit convention instead of silently mixing sequence offsets or combinatorial interpretations.

The result refines a Catalan total by peaks; it is not the count of all Dyck paths at that semilength. The two displayed binomial coefficients make the integer result reproducible. Recompute the displayed recurrence or closed form with the same inputs and compare its previous terms or counting factors. That check supports this bounded result, but it does not transfer the interpretation to a different sequence or counting object.

Semilength is capped at 30 so every accepted Narayana count stays exact. The cap preserves exact combinatorial multiplication. Combinatorial and recurrence values can grow rapidly even when the inputs look small. The conservative cap prevents browser stalls and avoids presenting an unsafe floating-point integer as exact; larger work needs arbitrary-precision software and independent resource controls.