A sequence of polynomials related to Catalan numbers The sequence of polynomials
$$P_n=\sum_{k=0}^{\lfloor(2n-1)/3\rfloor}
\frac{(2n-2k-1)!(2n-2k-2)!}{k!(n-k)!(n-k-1)!(2n-3k-1)!}x^k$$
satisfies apparently the identities
$$0=\sum_{j=0}^nP_{n-j}(P_j-(-x)^j)$$
for all $n\geq 2$. (The previous condition
$n\geq 1$ was incorrect, as pointed out in the answer of Ira Gessel.) (This has probably a WZ proof since it involves hypergeometric stuff, see the answers below.)
It is easy to see that $P_1,P_2,\ldots$ evaluates to the sequence $1,1,2,5,14,\ldots$ of Catalan numbers at $x=0$. Leading coefficients are also closely related to Catalan numbers and central binomial coefficients.
All roots of the polynomials $P_n$ are apparently in the real interval
$(-\infty,-16/27)$ and the
largest root of $P_n$ converges rather quickly (for $n\rightarrow \infty$) to the rational number $-16/27$.
Has this sequence of polynomials appeared elsewhere? Other interesting properties?
 A: I find a slightly different initial condition for the recurrence:
$$0=\sum_{j=0}^nP_{n-j}(P_j-(-x)^j)$$
for $n\ne 1$; for $n=1$ the sum is $-1$. It's easy to derive a formula for the generating function
$\sum_{n=0}^\infty P_n(x) z^n$ from this recurrence. We find that, as noted by Tewodros,
$$\sum_{n=0}^\infty P_n(x) z^n = \frac{1-\sqrt{4z(1+xz)^2}}{2(1+xz)}.$$
In terms of the Catalan number generating function $c(u) = (1-\sqrt{1-4u})/(2u)$, this is
$z(1+xz) c\bigl(z(1+xz)\bigr)$.
From this we can easily derive the OP's formula for $P_n(x)$ in the form
$$P_n(x) = \sum_i C_i \binom{2i+1}{n-i-1}x^{n-i-1},$$
where $C_i$ is the Catalan number $\frac{1}{i+1}\binom{2i}{i}$.
A: Just a remark regarding a recurrence for $P_n$ (found by the Wilf-Zeilberger methodology):
$$(n + 4)P_{n+4} + (nx - 4n + 4x - 10)P_{n+3} - 6(2n + 3)xP_{n+2} - 6(2n + 1)x^2P_{n+1} - 2(2n - 1)x^3P_n=0.$$
Another remark is this: we may write
$$P_n=\sum_{i=0}^{n-1}\binom{2i+1}i\binom{2i+1}{n-1-i}\frac{x^{n-1-i}}{2i+1} \qquad \text{or}$$
$$P_n=\sum_{j=0}^{n-1}\sum_{k=0}^j\binom{2k+2}{j-k}\binom{2k}k
\frac{(-1)^{n-1-j}x^{n-1-k}}{k+1}.$$
One more: a bi-variate generating function
$$\sum_{n\geq0}P_n(x)t^n
=\frac{1-\sqrt{1-4t(tx + 1)^2}}{2t(tx + 1)}.$$
