I posted the question https://math.stackexchange.com/questions/2799068/is-the-value-of-sum-limits-k-1%e2%88%9e-frac1c-kn-known before on mathstackexchange but realised that it might be more appropriate for mathoverflow after seeing the answer.

Is the value of the sum $a_n:=\sum\limits_{k=1}^{\infty}\dfrac{1}{(C_k)^n}$ known for $n \geq 1$, where $C_k= \dfrac{1}{k+1} \dbinom{2k}{k}$ are the Catalan numbers?

In the mathstackexchange thread it was shown that $a_1= 1+\frac{4\pi}{9\sqrt{3}}$ and that calculating $a_2$ might be more complicated.