Another look at Zagier's formula for multiple zeta values involving Hoffman elements

In this paper, we give an elementary account into Zagier's formula for multiple zeta values involving Hoffman elements. Our approach allows us to obtain direct proof in a special case via rational zeta series involving the coefficient $\zeta(2n)$. This formula plays an important role in proving Hoffman's conjecture which asserts that every multiple zeta value of weight $k$ can be expressed as a $\mathbb{Q}$-linear combinations of multiple zeta values of the same weight involving $2$'s and $3$'s. Also, using a similar hypergeometric argument via rational zeta series, we produce a new Zagier-type formula for the multiple special Hurwitz zeta values.