  In 1990, Backelin showed that the number of numerical semigroups with
Frobenius number $f$ approaches $C_i \cdot 2^{f/2}$ for constants $C_0$ and
$C_1$ depending on the parity of $f$. In this paper, we generalize this result
to semigroups of arbitrary depth by showing there are
$\lfloor{(q+1)^2/4}\rfloor^{f/(2q-2)+o(f)}$ semigroups with Frobenius number
$f$ and depth $q$. More generally, for fixed $q \geq 3$, we show that, given
$(q-1)m < f < qm$, the number of numerical semigroups with Frobenius number $f$
and multiplicity $m$ is\[\left(\left\lfloor \frac{(q+2)^2}{4}
\right\rfloor^{\alpha/2} \left \lfloor \frac{(q+1)^2}{4}
\right\rfloor^{(1-\alpha)/2}\right)^{m + o(m)}\] where $\alpha = f/m - (q-1)$.
Among other things, these results imply Backelin's result, strengthen bounds on
$C_i$, characterize the limiting distribution of multiplicity and genus with
respect to Frobenius number, and resolve a recent conjecture of Singhal on the
number of semigroups with fixed Frobenius number and maximal embedding
dimension.
