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:

$\displaystyle \sum_{n=0}^{\infty} a_n y^n=x$

Now, we know $\displaystyle a_n=\frac{1}{n!} \frac{d^n x}{dy^n}$

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

$\displaystyle y(x)=\sum_{n=0}^{\infty} b_n x^n$

with $\displaystyle b_n=\frac{1}{n!} \frac{d^n y}{dx^n}$

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

$\displaystyle y_n=\frac{1}{x_1} \frac{d}{dy} x_{n-1}$

where $\displaystyle y_n=\frac{d^n y}{dx^n}$ and $\displaystyle x_n=\frac{d^n x}{dy^n}$

and therefore if I know what each $\displaystyle a_n$ is, I can recursively calculate the coefficients $\displaystyle b_n$

Here's the first few:

$\displaystyle y_1=\frac{1}{x_1} $

$\displaystyle y_2=-\frac{x_2}{x_1^3}$

$\displaystyle y_3=\frac{3x^2-x_1 x^3}{x_1^5}$

$\displaystyle y_4=-\frac{15x_2^3-10 x_1 x_2 x_3+x_1^2 x_4}{x_1^7}$

$\displaystyle y_5=\frac{105 x_2^4-105 x_1 x_2^2 x_3+15x_1^2x_2 x^4+10 x_1^2x_3^2-x_1^3 x_5}{x_1^9}$

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

I can begin to see a trend for $\displaystyle n\geq 2$:

$\displaystyle y_n=(-1)^{n+1}\frac{\prod_{k=0}^{n-2} (2n+1) x_2^{n-1}+\text{something}+(-1)^{n+1}x_1^{n-2} x_n}{x_1^{2n-1}}$