I'm given the following recursive formula for a sequence:
$u_{n+1}=u_n+\sum_{k=1}^{n-1}u_ku_{n-k}\ (n\ge1),\ u_0=0,\ u_1=1$
Since $u_0=0$ we can rewrite:
$u_{n+1}=u_n+\sum_{k=0}^{n}u_ku_{n-k},\ (n\ge1)$
now writing $u(x)=u_1x+u_2x^2+....$ we obtain (by multiplying by $x^{n+1}$ and summing):
$u(x)-x=xu(x)+xu(x)^2$
Which I believe we can just solve using
$xu(x)^2+(x-1)u(x)+x=0$
So
$u(x)=\frac{1-x\pm\sqrt{(x-1)^2-4x^2}}{2x}$
However, my book says the solution should be
$u(x)=\frac{2x}{1-x+\sqrt{1-2x-3x^2}}$
Which is my answer to the power $-1$. I don't see why this is necessary, so any hints would be much appreciated.