For which values does the Gamma function yield an integer result?

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Gamma Function:

$$\Gamma(t)=\int_{0}^{\infty}x^{t-1}e^{-x}dx$$

Is it known for which values of $t$ (real or complex), the value of $\Gamma(t)$ is integer?

Are there any known specific patterns of $t$, for which the value of $\Gamma(t)$ is integer?