Some first term of this sequence.

Question

Let a generating function: $$(x^n A(x))' $$ How to determine some first term of this sequence. Thanks in advance.

Answer 1

Assuming $A(x) = \sum_{n=0}^\infty a_n n^n$ then

$$ (x^n A(x))' = nx^{n-1} x^n A(x) + x^n A'(x) = nx^{n-1} x^n A(x) + x^n \sum_{n=1}^\infty a_nnx^{n-1}$$