The motivation for this question is to find a closed-form representation for Catalan's constant. Formula (1) below for the Dirichlet beta function $\beta(s)$ (which I believe is globally convergent) leads to formulas (2), (3), and (4) below for Catalan's constant $G=\beta(2)$.
$$\beta(s)=\underset{K\to\infty}{\text{lim}}\left(\sum\limits_{n=0}^K\frac{1}{2^{n+1}}\sum\limits_{k=0}^n\frac{(-1)^k\ \binom{n}{k}}{(2 k+1)^s}\right)\tag{1}$$
$$G=\underset{K\to\infty}{\text{lim}}\left(\sum\limits_{n=0}^K\frac{1}{2^{n+1}}\sum\limits_{k=0}^n\frac{(-1)^k\ \binom{n}{k}}{(2 k+1)^2}\right)\tag{2}$$
$$G=\underset{K\to \infty }{\text{lim}}\left(\frac{\sqrt{\pi}}{8}\sum\limits_{n=0}^K\frac{n! \left(H_{n+\frac{1}{2}}+\log (4)\right)}{2^n\ \Gamma\left(n+\frac{3}{2}\right)}\right)\tag{3}$$
$$G=\underset{K\to \infty }{\text{lim}}\left(\frac{\sqrt{\pi}}{8}\left(\sqrt{\pi} \log (4)+\sum\limits_{n=0}^K \frac{n!\ H_{n+\frac{1}{2}}}{2^n\ \Gamma\left(n+\frac{3}{2}\right)}\right)\right)\tag{4}$$
Question: Is there a closed form representation for formula (5) below?
$$G=\underset{K\to \infty }{\text{lim}}\left(\sum\limits_{n=0}^K \frac{n!\ H_{n+\frac{1}{2}}}{2^n\ \Gamma\left(n+\frac{3}{2}\right)}\right)=1.6770831981...\tag{5}$$
I tried using the following identity for the Harmonic number function but this just led to a more complicated infinite series over Hypergeometric PFQ Regularized functions.
$$H_{n+\frac{1}{2}}=\sum _{m=1}^{\infty } \frac{n+\frac{1}{2}}{m \left(m+n+\frac{1}{2}\right)}\tag{6}$$