New series involving binomial coefficients (II)

In this paper, we evaluate some series of the form $$\sum _{k=1}^{\mathrm{\infty }}\frac{a{k}^2+bk+c}{k(3k-1)(3k-2)m^k\left(\genfrac{}{}{0px}{}{4k}{k}\right)}\mathrm{.}$$ For example, we prove that $$\sum _{k=1}^{\mathrm{\infty }}\frac{(5k^2-4k+1)8^k}{k(3k-1)(3k-2)\left(\genfrac{}{}{0px}{}{4k}{k}\right)}=\frac{3}{2}\pi$$ and $$\sum _{k=1}^{\mathrm{\infty }}\frac{415k^2-343k+62}{k(3k-1)(3k-2)(-8{)}^k\left(\genfrac{}{}{0px}{}{4k}{k}\right)}=-3\log 2.$$ We also pose many new conjectural series identities involving binomial coefficients; for example, we conjecture that $$\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\genfrac{}{}{0px}{}{2k}{k}\right)}^3}{{4096}^k}(9(42k+5)\sum _{0\le j<k}\frac{1}{(2j+1{)}^4}+\frac{25}{(2k+1{)}^3})=\frac{5}{6}{\pi }^3\mathrm{.}$$