I am looking at Bandeira–Boedihardjo–van Handel's definition of a free semicircular family, on p.24. Right below, they claim that $$\lvert NC_2(p)\rvert=\frac{1}{2\pi}\int_{-2}^2x^p\sqrt{4-x^2}dx$$ for even $p$. Why is this the case? I don't see how moments of this distribution should connect to this counting problem.
It seems that these are just the Catalan numbers (Mathematica agrees on the first several; Wikipedia claims it, though I can't tell if their $NC$ is the same as [BBvH]'s $NC_2$).
Go to Banica's Free Probability text here.
Consult Theorem 7.8 and Proposition 7.26.
I think Banica defines Catalan numbers thus, but then shows they satisfy the usual recurrence.