The number of ways to pile n white circles, when in the bottom row there are n white circles?

Question

I am struggling with this Question: What are the number of ways to pile circles (as long as they don't fall from the sides) when in the bottom row there are $n$ white circles. each row can hold maximum a number of one less circles than the row underneath. for example for $n=3$ the answer is 5.

i think it might be recurrence that we have $a_1=1$ and $a_2=2$ but i don't know what the formula can be.

Answer 1

Put one more row with $n+1$ balls underneath the arrangement. Start at the leftmost of these balls, and in each step, go right by half a ball width, staying on existing balls, moving up if there's a ball there and down if not. The resulting paths are precisely the Dyck paths of length $2n$, which are counted by the Catalan numbers.