  We study the topological string partition function of a class of toric,
double elliptically fibered Calabi-Yau threefolds $X_{N,M}$ at a generic point
in the K\"ahler moduli space. These manifolds engineer little string theories
in five dimensions or lower and are dual to stacks of M5-branes probing a
transverse orbifold singularity. Using the refined topological vertex
formalism, we explicitly calculate a generic building block which allows to
compute the topological string partition function of $X_{N,M}$ as a series
expansion in different K\"ahler parameters. Using this result we give further
explicit proof for a duality found previously in the literature, which relates
$X_{N,M}\sim X_{N',M'}$ for $NM=N'M'$ and $\text{gcd}(N,M)=\text{gcd}(N',M')$.
