Doubt on a recurrence equation I have the following recursive equation for a pmf $P(s)$:
$$P(s+1)=2 \lambda (1-\lambda) P(s)+ \lambda^2 \sum_{s'=0}^s P(s') P(s-s')$$
Here $s$ is a natural number and I have two initial conditions $P(0)=0$ and $P(1)=(1-\lambda)^2$.
I tried to solve it using a generating function $G(x)=\sum_{s=0}^\infty x^s P(s)$. Using it I get:
$$\frac{1}{x} G(x)=2 \lambda (1-\lambda) G(x) + \lambda^2 G^2(x)$$
but I think something is wrong (i.e. it does not satisfy $G(1)=1$). Any suggestions?

Cintuing Simply Beautiful Art's work.
$G(x)=(1-\lambda)^2x+2\lambda(1-\lambda)xG(x)+\lambda^2xG(x)^2
$
so,
writing $c$ for $\lambda$,
$0
=c^2xG(x)^2+(2c(1-c)x-1)G(x)+(1-c)^2x
$.
$\begin{array}\\
D
&=(2c(1-c)x-1)^2-4c^2x(1-c)^2x\\
&=4c^2(1-c)^2x^2 -4c(1-c)x+1-4c^2x(1-c)^2x\\
&=-4c(1-c)x+1\\
&=1+(4c^2-4c)x\\
&=1+(4c^2-4c+1-1)x\\
&=1-x+(2c-1)^2x\\
\end{array}
$
so
$\begin{array}\\
G(x)
&=\dfrac{-(2c(1-c)x-1)\pm\sqrt{1-x+(2c-1)^2x}}{2c^2x}\\
&=\dfrac{2c^2x-2c+1\pm\sqrt{1-x+(2c-1)^2x}}{2c^2x}\\
&=1-\dfrac{2c-1\mp\sqrt{1-x+(2c-1)^2x}}{2c^2x}\\
G(1)
&=1-\dfrac{2c-1\mp\sqrt{(2c-1)^2}}{2c^2}\\
&=1-\dfrac{2c-1\mp(2c-1)}{2c^2}\\
&=1-\dfrac{0, 4c-2}{2c^2}\\
&=1, 1-\dfrac{3c-1}{c^2}\\
\end{array}
$
So $G(1) = 1$
for the positive square root.
