generating function of $(1/n)a_n$ in terms of the generating function of $a_n$

Question

I have the generating function of $a_n$, $A(z)$, and I want to find the generating function of $(1/n)a_n$ in terms of the generating function of $a_n$. Any help is appreciated.

Answer 1

You are given \begin{align*} A(z) & =\sum_{n=0}^{\infty}a_nz^{n}\\ A(z)-a_0 & =a_1z+a_2z^2+\dotsb + a_nz^n +\dotsb\\ \frac{A(z)-a_0}{z} & =a_1+a_2z^1+\dotsb + a_nz^{n-1} +\dotsb\\ \int \frac{A(z)-a_0}{z} \, dz & =a_1z+\frac{a_2}{2}z^2+\dotsb + \frac{a_n}{n}z^n +\dotsb. \end{align*}

Thus the generating function is $$\int \frac{A(z)-a_0}{z} \, dz$$