I'm interested in the sum, $$\sum_{n=1}^\infty\frac{\zeta(2n)\Gamma(2n)}{\Gamma(2k+2n+2)}x^{2n}$$ Otherwise written as $$\sum_{n=1}^\infty\frac{\zeta(2n)}{(2n)(2n+1)\cdots(2n+2k+1)}x^{2n}$$
I am looking for some values of $k\gt0\in\mathbb{Z}$ and $x\ne0\in\mathbb{R}$ for which the expression equals a value in terms of known constants preferably in a closed form.