  Using cyclotomic multiple zeta values of level $8$, we confirm and generalize
several conjectural identities on infinite series with summands involving
$\binom{2k}k8^k/(\binom{3k}k\binom{6k}{3k})$. For example, we prove that
\[\sum_{k=0}^\infty\frac{(350k-17)\binom{2k}k8^k}
{\binom{3k}k\binom{6k}{3k}}=15\sqrt2\,\pi+27\] and
\[\sum_{k=1}^\infty\frac{\left\{(5k-1)\left[16\mathsf H_{2k-1}^{(2)}-3\mathsf
H_{k-1}^{(2)}\right]-\frac{12(6k-1)}{(2k-1)^2}\right\}\binom{2k}k8^k}
{k(2k-1)\binom{3k}k\binom{6k}{3k}}=\frac{\pi^3}{12\sqrt2},\] where $\mathsf
H^{(2)}_m$ denotes the second-order harmonic number $\sum_{0<j\leq
m}\frac1{j^2}$.
