  A power series is called lacunary if `almost all' of its coefficients are
zero. Integer partitions have motivated the classification of lacunary
specializations of Han's extension of the Nekrasov-Okounkov formula. More
precisely, we consider the modular forms \[F_{a,b,c}(z) := \frac{\eta(24az)^a
\eta(24acz)^{b-a}}{\eta(24z)},\] defined in terms of the Dedekind
$\eta$-function, for integers $a,c \geq 1$ where $b \geq 1$ is odd throughout.
Serre determined the lacunarity of the series when $a = c = 1$. Later, Clader,
Kemper, and Wage extended this result by allowing $a$ to be general, and
completely classified the $F_{a,b,1}(z)$ which are lacunary. Here, we consider
all $c$ and show that for ${a \in \{1,2,3\}}$, there are infinite families of
lacunary series. However, for $a \geq 4$, we show that there are finitely many
triples $(a,b,c)$ such that $F_{a,b,c}(z)$ is lacunary. In particular, if $a
\geq 4$, $b \geq 7$, and $c \geq 2$, then $F_{a,b,c}(z)$ is not lacunary.
Underlying this result is the proof the $t$-core partition conjecture proved by
Granville and Ono.
