I don't remember any method to compute the closed from for the following series. $$ \sum_{k=0}^{\infty}\binom{3k}{k} x^k .$$
I tried by putting $\binom{3k}{k}$ in Mathematica for different $k$ and asking for the generating function it deliver a complicated formula which is the following. $$ \frac{2\cos[\frac{1}{3} \sin^{-1}(\frac{\sqrt{3x}}{2})]}{\sqrt{4-27x}} $$
I was wondering if there is any simple form?