The result in the OP
$$\sum_{n=1}^\infty \sum_{k=1}^\infty\frac{G_n}{k}\frac{\Gamma(k+\frac{3}{2})\Gamma(n+\frac{1}{2})}{(n+k+1)!}=\frac{\pi(1-8\log 2)}{8}=-1.78489,$$ does not seem to agree with a numerical estimate of the series. I first evaluated the sum over $k$, which has a closed form expression, $$\sigma_n=\sum _{k=1}^{\infty } \frac{\Gamma \left(k+\frac{3}{2}\right) \Gamma \left(n+\frac{1}{2}\right)}{k (k+n+1)!}=\frac{\sqrt{\pi }\, \Gamma \left(n+\frac{1}{2}\right) \left(\psi ^{(0)}(n+2)-\psi ^{(0)}\left(n+\frac{1}{2}\right)\right)}{2 \Gamma (n+2)},$$ with $\psi^{(0)}$ the polygamma function, and then evaluated $$S_N=\sum_{n=1}^N G_n \sigma_n,$$ using the following Mathematica code for the Gregory coefficients:
g[n_] := Integrate[x*Pochhammer[x - n + 1, n - 1]/n!, {x, 0, 1}]
The plot of $S_N$ versus $N$ below suggests convergence to 0.1665, very far from the result in the OP.
