Multiple Clausen values and deformed Ap\'ery-like series

With generalized central binomial coefficients $ \binom{2x}{x}:=\frac{\Gamma(2x+1)}{[\Gamma(x+1)]^2}$ defined through Euler's gamma function, we represent deformed Apéry-like series \[ \mathscr A_{s,n}:=\sum_{k=1}^\infty\left.\!\frac{\partial^n}{\partial x^n}\frac{1}{x^s\binom{2x}{x}}\right|_{x=k} \] by multiple Clausen values (MCVs), which belong to a special class of cyclotomic multiple zeta values (CMZVs) at level $3$.

For example, exploiting provable algebraic relations among MCVs, we show that \[\mathscr A_{1,5}=-\frac{9[495L(\chi_{-3},6)-30\pi^{2}L(\chi_{-3},4)-2\pi^{4}L(\chi_{-3},2)]}{4}\]and\[\mathscr A_{4,4}=\frac{352\zeta_{5,3}}{15}+\frac{752537\pi^{8}}{10206000},\]where $ L(\chi_{-3},s):=\sum_{n=0}^\infty\left[(3n+1)^{-s}-(3n+2)^{-s}\right]$ and $ \zeta_{5,3}:=\sum_{m=1}^\infty\sum_{n=1}^{m-1}m^{-5}n^{-3}$.