Our goal is evaluating: $$S=\sum_{n=0}^\infty(in)!= \sum\limits_{n=0}^\infty\Gamma(in+1) = 1.66159412484058… - 0.09593230517926 …i $$ which hypergeometric function generalizations cannot express. Attempting to rewrite it using $\Gamma(n)$’s integral representation does not work either as the geometric series converges for all $0\le x$. Even using a regularization and Cauchy principal value at $x=1$: $$S= \sum_{n=0}^\infty\int_0^\infty e^{-x}x^{i n}dx\mathop=^\text{reg}\int_0^\infty\frac{e^{-x}}{1-x^i}dx\mathop\approx^\text{PV}0.5-0.041380i$$ gives incorrect numerical results. The last idea was to convert it to a Barnes type integral, like with a Ramanujan interpolation based formula. With $\Gamma(in+1)$, the result diverges, but using $\Gamma(1-in)$:
$$\bar S=1+\frac i2\int_{c-i\infty}^{c+i\infty}((\cot(\pi z)+i)\Gamma(1-i z)dz\iff S=1-\frac i2\int_{ci-\infty}^{ci+\infty}(\coth(\pi z)-1)\Gamma(z+1)dz$$
Another useful formula may be $\Gamma(1-in)=-\pi i\frac{\operatorname{csch}(\pi n)}{\Gamma(i n)}$. Finally, Since $\lim_\limits{n\to\infty}\frac{2\ln(\text{Re}(\Gamma(in+1)))}{\pi n}=-1$, the sum’s real part roughly converges at $e^{-\frac\pi2n}$ speed. However, not much so far helps evaluate the sum.
What can we do?
Consider the power series $$ F(z)=\sum_{n\geq 0}z^n\Gamma(i n+1). $$ Since $|\Gamma(i n+1)|\sim \sqrt{2\pi n}\,e^{-\frac \pi 2n}$ as $n\to+\infty$ (cf. DLMF), this defines a function analytic for $|z|<e^{\pi/2}$. In particular, $F(1)$ is the value we want to compute.
The following facts should be true.
For $|z|<1$ we have the integral representation $$ F(z)=\int_0^{+\infty}\frac{e^{-s}}{1-zs^i}ds. $$
We have $$ F(1)=\lim_{z\to 1_-}\int_0^{+\infty}\frac{e^{-s}}{1-zs^i}ds $$ where the notation indicates that the limit is taken as $z\to 1$ with $|z|<1$.
This limit can be computed by the Sokhotski$-$Plemelji formula to be $$ F(1)=PV\int_0^{+\infty}\frac{e^{-s}}{1-s^i}ds+\pi \sum_{n\in\mathbb Z}\exp\bigl(-\exp(2\pi n)\bigr)\exp(2\pi n). $$