  In 2008, Han rediscovered an expansion of powers of Dedekind $\eta$ function
attributed to Nekrasov and Okounkov (which was actually first proved the same
year by Westbury) by using a famous identity of Macdonald in the framework of
type $\widetilde{A}$ affine root systems. In this paper, we obtain new
combinatorial expansions of powers of $\eta$, in terms of partition hook
lengths, by using the Macdonald identity in type $\widetilde{C}$ and a new
bijection between vectors with integral coordinates and a subset of $t$-cores
for integer partitions. As applications, we derive a symplectic hook formula
and an unexpected relation between the Macdonald identities in types
$\widetilde{C}$, $\widetilde{B}$, and $\widetilde{BC}$. We also generalize
these expansions through the Littlewood decomposition and deduce in particular
many new weighted generating functions for subsets of integer partitions and
refinements of hook formulas.
