Compute central Delannoy number using Snake Oil I'm stuck doing this problem on central Delannoy numbers: starting with $ d_{n,n} = \sum_k \binom{n+k}{2k} \binom{2k}{k} $, I have to use Snake Oil to find $ \sum d_{n,n} x^n $. After exchanging the variables I get $$ \sum_k \binom{2k}{k} x^k \sum_n \binom{n+2k}{2k} x^n = \sum_k \binom{2k}{k} \frac{x^k}{(1-x)^{2k+1}} $$.
I don't know what to do next at this point. How do I simplify this sum?

You can use the well-known $$\sum^\infty_{k=0}\binom{2k}{k}y^k=\frac1{\sqrt{1-4y}},$$ set $$y=\frac{x}{(1-x)^2}$$ and divide by $1-x$ to arrive at the generating function
$$\frac1{\sqrt{1-6x+x^2}}.$$
