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

Theorem
Let $\sequence {a_n}$ be the sequence whose terms are defined as:
 * $\forall n \in \Z_{\ge 0}: a_n = \ds \sum_{k \mathop = 1}^{n - 1} \dfrac 1 {k \paren {n - k} }$

Then $\sequence {a_n}$ has the generating function $\map G z$ such that:


 * $\map G z = \paren {\ln \dfrac 1 {1 - z} }^2$

and whose terms are:


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

Proof
From Product of Generating Functions:


 * $\map G z = \paren {\map {G_1} z}^2$

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

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

Hence:


 * $\map G z = \paren {\ln \dfrac 1 {1 - z} }^2$

Differentiating $\map G z$ $z$ gives:

Integrating again $z$ gives: