It is well known that
$$\displaystyle \mathcal{H}_n \sim \log{n} + \gamma - \sum_{k=1}^\infty \frac{B_{k}}{k \, n^{k}},$$
where $\displaystyle B_k$ are the Bernoulli numbers.
A similar asymptotic expansion for the Harmonic numbers begins with the logarithm shifted slightly:
$$\displaystyle \mathcal{H}_n = \log{\left(n+\frac{1}{2}\right)} + \gamma + \sum_{k=2}^\infty \frac{A_k}{n^k}$$
Is there a closed form for $\displaystyle A_k$ in terms of known number sequences?
The first few terms of $\displaystyle A_k$ are:
$$\displaystyle \frac{1}{24}, -\frac{1}{24}, \frac{23}{960}, -\frac{1}{160}, -\frac{11}{8064}, -\frac{1}{896}, \frac{143}{30720}, -\frac{1}{4608}$$
for $\displaystyle k \in \left[2,\ldots,9 \right]$.
Simply write
$$\begin{align} \log\left(n+\frac12\right)&=\log(n)+\log\left(1+\frac1{2n}\right)\\\\ &=\log(n)+\sum_{k=1}^\infty \frac{(-1)^{k-1}}{2^kk\,n^k} \end{align}$$