I have a monotonic polynomial recurrence of the following form: c_n = 1-p + p*(c_n-1)^2

This comes from the probability that a specific branching process (Galton-Watson) will be extinct before the nth generation. It's generalization is:

c_n = 1-p_1-p_2-...-p_n + p_1(c_n-1) + p_2(c_n-1)^2 + ... + p_n(c_n-1)^n

Where p_1*1+p_2*2+...+p_n*n < 1, and c_1 << 1;

I know that polynomial recurrences have no general solution, but the monotonic case seems like it should be MUCH easier. A basic approximation for my specific c_n is easily done: d_n = 1-c_n =1- (1-p + p*(1-c_n-1)^2) d_n = 2p*d_n-1 - p*(d_n-1)^2 < 2p*(d_n-1) = (2p)^(n-1)*(d_1);

A cobweb plot also gives strong indication of the behavior.

My problem is, however, that although asymptotic behavior is easy to figure out, I need to be able to determine, with a fair degree of accuracy, d_i for arbitrary i.

Is there any information out there on monotonic recurrences of polynomial form?

Update: Any sort of approximation that's more general and useful than mine is also useful to me. Also, I'm only interested in real numbers. For instance, d_1 in the specific case I'm looking at is p, where p is < 0.5. Looking at cobweb plot with y = x and y = 2p*x - p*x^2 = px(2 - x), it is clear that the recurrence is monotonic and has no chaotic behavior. Is this not a strong enough of a condition to give it a (somewhat) clean form? If so, why not?