I'm trying to solve the recurrence $$ a_j = 0 \quad j \leq 0\\ a_1 = 1\\ \sum_{k=0}^{\frac{n}{2}}a_ka_{n-k} = 2\sum_{k=0}^{\frac{n}{2}}a_ka_{n-2k} - \sum_{k=0}^{\frac{n}{2}}a_ka_{n-4k} \qquad n \geq 2 $$
The equation I got for the generating function$$A(x) = \sum_{n=0}^{\infty}a_nx^n$$ $${A(x)}^2 = (2A(x^2) - A(x^4))(A(x)-A(-x)) + 2A(x^4)(A(x^2) - A(-x^2))$$ This level of hideousness is beyond my capabilities....and wolfram alpha's. How can I solve for A(x)?