I am interested in the behaviour, as $n$ is great, of the coefficients $g_n$ in the Maclauren expansion of $\displaystyle \frac{1}{\Gamma(1+x)} $.
We have $$ \frac{1}{\Gamma(1+x)}=\sum_{n=0}^\infty g_n x^n $$ with $$ \begin{align}g_0 &=1 \\g_1 &=\gamma \\g_2 &=\frac{\gamma^2}{2}-\frac{\pi^2}{12} \\g_3 &=\frac{\gamma^3}{6}-\frac{\pi^2\gamma}{12}+\frac{\zeta(3)}{3} \\... \end{align} $$ and $g_n \rightarrow 0.$
I would appreciate any asymptotic expansion giving the behavior of $g_n$ as $n$ tends to $+\infty$.
The following recent arXiv preprint (uploaded less than 3 weeks ago!) explicitly discusses the Taylor series expansions of the reciprocal Gamma function, and derives asymptotics. To do so, the author expresses the coefficients $g_n$ (labelled as $a_n$ in the paper---see equation $2.2$) using the Cauchy integral formula, and then does a saddle-point analysis. This yields
$$g_n\sim (-1)^n \sqrt{\frac{2}{\pi}}\frac{\sqrt{n}}{n!}\text{Im}\left\{e^{-n z_0}\frac{z_0^{1/2-n}}{\sqrt{1+z_0}}\right\}\tag{eq. 3.13}$$
where $z_0=-n^{-1}\exp(W_{-1}(-n))$ is the saddle point ($W_{-1}$ is a branch of the Lambert $W$ function.) He also cites a classical upper bound due to Bouroget $$g_n\leq \frac{(-1)^n}{\pi n!}\frac{e \pi^{n+1}}{n+1}+\frac{4}{\pi^2}{\sqrt{n!}} \lesssim \frac{4}{\pi^2\sqrt{n!}}\tag{eq. 1.5}$$ and a different saddle-point formula derived via Heyman's asymptotic formula (eqs. 3.14-3.16). The papers cited throughout the preprint would likely contain valuable information as well.