The following is a purely computational; including a process of converting a generating function to its formal power series relations. This can be also seen as an example of recurrence relation at the 2nd-order with indices coefficient.
Observe a case of
To determine the , notice that the k-derivative has a form
where and is a monic polynomial of degree (k-1) with .
By a calculation, one can see has an inductive formula
Combining these formula, we have
First few and are as follows.
By close look at the first-order term of , holds. Thus there we have recurrence relation determined solely by means of itself