How to show that $\sum_{n=1}^{\infty}\frac{\phi^{2n}}{n^2{2n \choose n}}=\frac{9}{50}\pi^2$ Given:$$\sum_{n=1}^{\infty}\frac{\phi^{2n}}{n^2{2n \choose n}}=\frac{9}{50}\pi^2$$
Where $\phi=\frac{\sqrt{5}+1}{2}$
How can I we show that the above sum is correct?
I have checked numerically, it seem correct, but i don't how to proves it 

Replace $\phi$ with $x$. That gives a function of $x$ with its Taylor series.
Now to find a differential equation that the Taylor series obeys.
Hint:  
On one hand, $d^2f/dx^2=\sum (2n+2)(2n+1)a_{n+1}x^{2n}$. 
On the other hand $xdf/dx=\sum 2n a_n x^{2n}$.  
Good luck with that.  As a second hint, other people have supplied the answer.
