If one can solve the differential equation $$z^2(z-16)f''(z)+5z^2f'(z)+4(z-1)f(z)+4=0$$ with initial conditions $f(0)=1$ and $f'(0)=1$ then $$a_2=f(1).$$

Let $f(z)=\sum_{n=0}^{\infty}\frac{z^n}{C_n^2}$ is a generating function then it satisfies the differential equation $$z^2(z-16)f''(z)+5z^2f'(z)+4(z-1)f(z)+4=0$$ with initial conditions $f(0)=1$ and $f'(0)=1$. Solving this leads to $$a_2=f(1).$$

If one can solve the differential equation $$z^2(z-16)f''(z)+5z^2f'(z)+4(z-1)+4=0$$ with initial conditions $f(0)=1$ and $f'(0)=1$ then $$a_2=f(1).$$

If one can solve the differential equation $$z^2(z-16)f''(z)+5z^2f'(z)+4(z-1)f(z)+4=0$$ with initial conditions $f(0)=1$ and $f'(0)=1$ then $$a_2=f(1).$$