I'm having a bit of trouble with this proof and was wondering if I could get some assistance. I need to show that: $$ΦS(x) = \frac{1}{\sqrt{1-4x}}$$ All I have is: $$\sum\limits_{k=0}^n {2n\choose k}{2(n-k)\choose n-k} = 4^n$$
I know by definition that $$ΦS(x) = \sum\limits_{n\ge0}s_n\cdot x^n$$ the exponent is the weight and the coefficient is the number of elements with that weight. If I could get some help that would be great!