  We find two combinatorial identities on the theta series of the root lattices
of the finite-dimensional simple Lie algebras of type $D_{4n}$ and the cosets
in their integral duals, in terms of the well-known Essenstein series $E_4(z)$
and Ramanujan series $\Dlt_{24}(z)$. Using these two identities, we determine
the theta series of certain infinite families of postive definite even
unimodular lattices obtained by gluing finite copies of the root lattices of
the finite-dimensional simple Lie algebras of type
  $D_{2n}$. It turns out that these theta series are weighted symmetric
polynomials of two fixed families of polynomials of $E_4(z)$ and
$\Dlt_{24}(z)$.
