Multiple Clausen values and deformed Apéry-like series

With generalized central binomial coefficients $\left(\genfrac{}{}{0px}{}{2x}{x}\right):=\frac{\mathrm{\Gamma }(2x+1)}{[\mathrm{\Gamma }(x+1){]}^2}$ defined through Euler’s gamma function, we represent deformed Apéry-like series $${\mathcal{A}}_{s,n}:=\sum _{k=1}^{\mathrm{\infty }}{\text{\hspace{0.17em}⁣}\frac{{\mathrm{\partial }}^n}{\mathrm{\partial }{x}^n}\frac{1}{x^s\left(\genfrac{}{}{0px}{}{2x}{x}\right)}\mid }_{x=k}$$ 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 $${\mathcal{A}}_{1,5}=-\frac{9[495L({\chi }_{-3},6)-30{\pi }^{2}L({\chi }_{-3},4)-2{\pi }^{4}L({\chi }_{-3},2)]}{4}$$and$${\mathcal{A}}_{4,4}=\frac{352{\zeta }_{5,3}}{15}+\frac{752537{\pi }^8}{10206000},$$where $L({\chi }_{-3},s):={\sum }_{n=0}^{\mathrm{\infty }}[(3n+1{)}^{-s}-(3n+2{)}^{-s}]$ and ${\zeta }_{5,3}:={\sum }_{m=1}^{\mathrm{\infty }}{\sum }_{n=1}^{m-1}{m}^{-5}{n}^{-3}$.