Generating Function for Sequence of Sum over k to n of Reciprocal of k by n-k

Theorem
Let $\left\langle{a_n}\right\rangle$ be the sequence whose terms are defined as:
 * $\forall n \in \Z_{\ge 0}: a_n = \displaystyle \sum_{k \mathop = 1}^{n - 1} \dfrac 1 {k \left({n - k}\right)}$

Then $\left\langle{a_n}\right\rangle$ has the generating function $G \left({z}\right)$ such that:


 * $G \left({z}\right) = \left({\ln \dfrac 1 {1 - z} }\right)^2$

and whose terms are:


 * $a_n = \dfrac {2 H_{n - 1} } n$

Proof
From Product of Generating Functions:


 * $G \left({z}\right) = \left({G_1 \left({z}\right)}\right)^2$

where $G_1 \left({z}\right)$ is the generating function for $\displaystyle \sum_{k \mathop \ge 1} \dfrac 1 k$.

From Generating Function for Sequence of Reciprocals of Natural Numbers:
 * $G_1 \left({z}\right) = \ln \left({\dfrac 1 {1 - z} }\right)$

Hence:


 * $G \left({z}\right) = \left({\ln \dfrac 1 {1 - z} }\right)^2$

Differentiating $G \left({z}\right)$ $z$ gives:

Integrating again $z$ gives: