Note that only satisfies the ODE, not the initial conditions (1, 2).

To generally solve this, consider

so

and

Now put those two in the ODE ( ) and put them together under the same summation (consider the change of variable k = n-2 for y'')

You'll get a recursive formula for , from which you will be able to derive a general solution. After that, plug in the initial conditions to get the specific solution.