Assume that the differential equation
has a power-series solution and find a formula for the coefficient .
First, we have
Therefore, we have
Equating like powers of we obtain the recursive relation
By induction, we then have
It should be alpha^n/n!, at least. But how did you get that a_0 = 1?
To me it seems like it can be any constant so the solution is actually a0*alpha^n/n!.