I'm interested in inverting a power series: Suppose through solving a DE for the function y(x), I ended up with a power series in y:

Now, we know

I'm just learning I can invert that series to obtain:

with

But each derivative in y is related to each derivative in x by the recursive formula:

where and

and therefore if I know what each is, I can recursively calculate the coefficients

Here's the first few:

You guys think it's possible to figure out a non-recursive formula for ?

I can begin to see a trend for :