  Dyonic 1/4-BPS states in Type IIB string theory compactified on $\mathrm{K}3
\times T^2$ are counted by meromorphic Jacobi forms. The finite parts of these
functions, which are mixed mock Jacobi forms, account for the degeneracy of
states stable throughout the moduli space of the compactification. In this
paper, we obtain an exact asymptotic expansion for their Fourier coefficients,
refining the Hardy-Ramanujan-Littlewood circle method to deal with their
mixed-mock character. The result is compared to a low-energy supergravity
computation of the exact entropy of extremal dyonic 1/4-BPS single-centered
black holes, obtained by applying supersymmetric localization techniques to the
quantum entropy function.
