For each complex number $c$, define $P_{0}(c)=0$ and $P_{n+1}(c) = (P_{n}(c))^{2} + c$ . The Mandelbrot set is the set of complex numbers c for which $|P_{n}(c)|$ stays bounded as $n\rightarrow \infty$.

$P_{n}(c)$ is a polynomial in $c$. Donald D. Cross noticed in 2005 that the leading terms settle down

$p_{0} + p_{1}c + p_{2} c^{2} + p_{3} c^{3} + ... = c + c^{2} + 2c^{3} + 5c^{4} + ...$ and submitted them to the Online Encyclopedia of Integer Sequences (A000108). It was subsequently noticed that these are the terms for the generating function $P(c)$ of the Catalan numbers, which is defined by: $P(c) = cP(c)^{2} + 1$. Indeed, $p_{n} = \sum_{i=1}^{n-1}p_{i}p_{n-i}$.

Philippe Flajolet and Robert Sedgewick mention this connection with enthusiasm in their definitive textbook Analytic Combinatorics, 2009, available online for free. See pages 535-537 and also the link to the theta function is discussed on page 328-330. Curiously, there is no mention of the Mandelbrot set in Richard Stanley's 2015 book Catalan Numbers.

My question is whether $P(c)$ can be used to define the Mandelbrot set. Is it the case that $|P(c)|$ is bounded if and only if $|P_{n}(c)|$ stays bounded as $n\rightarrow \infty$ ? Is the Mandelbrot set simply the set for which $P(c)$ is bounded?