  The Hodge numbers of the Hilbert schemes of points on algebraic surfaces are
given by G\"ottsche's formula, which expresses the generating functions of the
Hodge numbers in terms of theta and eta functions. We specialize in this paper
to generating functions of the $\chi_y(\mathrm{K3}^{[n]})$ genera of Hilbert
schemes of $n$ points on K3 surfaces. We determine asymptotic values of the
coefficients of the $\chi_y$-genus for $n\to \infty$ as well as their
asymptotic profile.
