Extracting coefficients of a series counting directed animals of the lattice For a fixed integer $k$, consider the function :
$$f(z)= \frac{1}{2} \left(\sqrt{\frac{1+z}{1-3z}}-1 \right) \left(\frac{1-z-\sqrt{(1+z)(1-3z)}}{2z} \right)^k$$
I'd like to extract the $n$-th coefficient ;
$$a_n = [z^n] (f)$$
of $f(z)= \sum_{n \geq 0} a_n z^n$ , more precisely I am interested in an equivalent for $a_n$, not the exact formula; I presume the asymptotic will take the form:
$$a_n \sim \alpha_k \cdot 3^n, \qquad \text{ as } n \to \infty$$
where $\alpha_k$ depends on $k$ only.
Context:
$a_n$ counts the number of directed animals of the lattice $\mathbb Z^2$ with $n$ vertices and  a "compact source" of size $k$, see Flajolet and Sedgewick, Analytic combinatorics, Example 1.18.

Flajolet and Sedgewick also explains how to do this kind of asymptotic analysis; Corollary VI.1 suffices here, and see also Example VI.3 and VI.11. $f(z)$ has
dominant singularity $z = \frac{1}{3}$ and near this singularity behaves like
$$f(z) \sim \frac{1}{2} \left( \sqrt{ \frac{ \frac{4}{3} }{1 - 3z }} - 1 \right) \left( 1 - \sqrt{ 3 (1 - 3z)} \right)^k.$$
Then, if I am not horribly mistaken, the dominant term here is actually the very first one, proportional to $(1 - 3z)^{- \frac{1}{2} }$, so in fact
$$f(z) \sim \frac{1}{\sqrt{3(1 - 3z)}}$$
which gives
$$\boxed{ a_n \sim \frac{3^n}{\sqrt{3 \pi n}} }$$
independent of $k$. Again if I am not horribly mistaken, $k$ turns out only to affect lower-order terms in the asymptotic expansion.
