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?
Best Answer
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 $B\approx2311.3945621\dots\,$.