We have the $j$-invariant defined as


I have that
$$
j(\tau)=\frac{1}{q}+\sum\_{k\geq 0}c\_kq^k,
$$
where $q=e^{-2\pi t}$ ($\tau=it$) and $c\_k\sim e^{4\pi\sqrt{k}}/(k^{3/4}\sqrt{2})$.


The inversion formula for the $j$-invariant is
$$
q=j^{-1}+\sum\_{k\geq 2}d\_kj^{-k}.
$$


Thus, I would like to know some upper bound or asymptotic formula for $d\_k$.


Any hint or reference?



Growth of the coefficients of the inversion of the $j$-invariant function
It's in the OEIS: <https://oeis.org/A066396>


There's a formula there that gives an approximation of the form (in your notation)
$$
d\_k \sim A \cdot (-1)^{k+1}\cdot B^k / k^{3/2}
$$
where $A\approx1943.54943\dots$
and $B\approx2311.3945621\dots\,$.


