Find a closed form of the generating function
$$ y = x + \frac 23 x^3 + \frac 23 \frac 45 x^5 + \frac 23 \frac 45 \frac 67 x^7 + \cdots$$
I realized that $$y = (x+x^3+\cdots) - \frac 13 (x^3+x^5+\cdots) - \frac 23 \frac 15 (x^5+\cdots) - \cdots$$
$$(1-x^2)y = x - \frac 13 x^3 - \frac 23 \frac 15 x^5 - \cdots$$
$$((1-x^2)y)' = 1 - x^2 - \frac 23 x^4 - \cdots$$
$$1-((1-x^2)y)' = x^2 + \frac 23 x^4 + \cdots = x(x+\frac 23 x^3+\cdots) = xy$$
$$1-(-2x)y - (1-x^2)y' = xy$$
$$-(1-x^2)y' + xy + 1 = 0$$
I tried substituting $y=e^z$, but now I'm stuck.
