Series Solution to Second Order Linear ODE

Hey I had to miss 2 lectures this week due to illness, trying to catch up. Hoping someone can help me with my snag on a problem here, not sure how to deal with 3 different coefficient terms when working with the recursion formula... Thanks! I imagine I'm going wrong somewhere here...

So I started by differentiating the power series and substituting into the ODE.

Pulling the and terms into the sums.

So now we need to combine the sums into a single summation so we can equate coefficients for the recursion formula. We need to shift indexes and get the same power of .

So now we can add coefficients and combine the summations.

Which when we equate coefficients we obtain the following recursion formula;

This is where I hit a wall, I've tried plugging increasing values of n in to try to find something, but I am having trouble seeing any sort of pattern or where to go from here... I imagine i've made a mistake somewhere.

Just a push in the right direction would be great, thanks so much!

Kasper