How to find the number of increasing lattice paths from $(0,0)$ to $(n,n)$ that never cross but may touch the main diagonal (for example the line joining $(0,0)$ with $(n,n)$). I've found a similar question here (I did not manage to find it again) but I guess this case is a little different (because the final answer is $2c_n$, however in the question I saw it was $c_n$). Thanks for clarifying.
2026-03-25 15:53:40.1774454020
How to find the number of lattice paths
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JMoravitz's comment answers this. Turning that into an answer for completeness:
The Catalan numbers count various things including paths which stay on or below the diagonal (i.e. do not go above it): a previous question here asked for a proof.
The number of paths which stay on or above the diagonal is clearly the same number, and for $n \ge 1$ a path must go above or below the diagonal at the start so the combined number of paths which do not cross the diagonal is double the Catalan numbers.
OEIS A068875 gives the values for small $n$.