Another proof of Zagier's evaluation formula of the multiple zeta values $\zeta (2,\mathrm{.}\mathrm{.}\mathrm{.},2,3,2,\mathrm{.}\mathrm{.}\mathrm{.},2)$

Using some transformation formulas of the generalized hypergeometric series ${}_3{F}_2$, we give another proof of D. Zagier’s evaluation formula of the multiple zeta values $\zeta (2,\mathrm{.}\mathrm{.}\mathrm{.},2,3,2,\mathrm{.}\mathrm{.}\mathrm{.},2)$.