  In 2008, Han rediscovered an expansion of powers of Dedekind $\eta$ function
due to Nekrasov and Okounkov by using Macdonald's identity in type
$\widetilde{A}$. In this paper, we obtain new combinatorial expansions of
powers of $\eta$, in terms of partition hook lengths, by using Macdonald's
identity in type $\widetilde{C}$ and a new bijection. As applications, we
derive a symplectic hook formula and a relation between Macdonald's identities
in types $\widetilde{C}$, $\widetilde{B}$, and $\widetilde{BC}$.
