I am trying to solve identity involving binomials and Fibonacci numbers by using generating functions: $$\sum_{k=0}^n{n \choose k}{n+k\choose k}f_{k+1}=\sum_{k=0}^n{n \choose k}{n+k\choose k}(-1)^{n-k}f_{2k+1}$$
My computational approach by Mathematica lead me to derive this generating function:
$$\frac{\sqrt{3x^2-2x+3+2\sqrt{x^4-8x^3-2x^2-8x+1}}}{\sqrt{5}\sqrt{x^4-8x^3-2x^2-8x+1}}$$
Can someone show how to transform both or any of the identity sides to obtain (coefficiens of) this generating function.