According to Donald D. Cross, the characteristic polynomial of the $n^{\mathrm{th}}$ iteration of the Mandelbrot function, $z_{n+1} = z_n^2 + c$ where $z_0 = 0$, gives the Catalan numbers as the polynomial coefficients as $n \to \infty$.
For example, expanding $z_3$ gives you the coefficients $\{1, 1, 2, 1\}$, or $1c^1 + 1c^2 + 2c^3 + 1c^4$. With each iteration, the detail of the Mandelbrot boundary contour gets more detailed. Images below describe this, with the contour plot $|f(z)| =2$:
This is the plot for iteration 4, $\{1, 1, 2, 5, 6, 6, 4, 1\}$:
This is the plot for iteration 6, (see Cross' webpage for co-efficients):
An issue arrives when you attempt to find the summation equivalent to having Catalan co-efficients:$$\sum_{n=0}^{\infty} C_n x^{n+1} = x \sum_{n=0}^{\infty} C_n x^n.$$
If you use the generating function for the Catalan co-efficents we get the function: $\frac{1-\sqrt{1-4x}}2$. Which very obviously is not a detailed representation of the Mandelbrot boundary contour. Image below:
This is obviously not the curve I am looking for, so what am I missing?
You can use a Laurent series to approximate the equipotentials (level sets) of the Mandelbrot set [1]:
$$ \psi(w) = w + \sum_{n=0}^{\infty} \frac{b_n}{w^n} $$
This is an explicit conformal map from the exterior of the unit disk to the exterior of the Mandelbrot set, with a recursive formula for the coefficients $b_n$. See also OEIS 3, 4.
Setting $w = r e^{2 \pi i t}$ and letting $r \to 1$ from above, gives a parameterized family of curves $w_r(t)$ that might converge to the boundary of the Mandelbrot set (some things might obstruct this, I think convergence might depend on whether "all (including irrational) external rays land" or "Mandelbrot set is locally connected" or some such, I'm not sure..).
But, practically, it is very slow to converge, so other methods may be better.
[1]: http://www.mrob.com/pub/muency/laurentseries.html , which cites [2]
[2]: John H. Ewing and Glenn Schober, The area of the Mandelbrot set, Numer. Math., vol. 61 (1992), pages 59-72.