Sum of Product of Catalan Numbers

Question

Consider all arrays in the form of $x_1, x_2, ... x_k $ such that the sum of these numbers equal L. It is wanted the sum of all $C(x_1) * C(x_2) * ... * C(x_k)$ where $C(i)$ is i-th Catalan number. Is there any formula to calculate such value?

Answer 1

The generating function for Catalan numbers is $$ g(z) = \sum_{n\geq 0} C_n z^n = \sum_{n\geq 0}\binom{2n}{n}\frac{z^n}{n+1}=\frac{1-\sqrt{1-4z}}{2z}\tag{A}$$ and the number you want is just the coefficient of $z^L$ in $\left(\frac{1-\sqrt{1-4z}}{2z}\right)^k$, i.e.

$$ \frac{1}{2^k}[z^{L+k}]\left(1-\sqrt{1-4z}\right)^k = 2^{2L+k} [z^{L+k}]\left(1-\sqrt{1-z}\right)^k\tag{B}$$

By Lagrange inversion Theorem, this number equals $\color{blue}{\frac{k}{2L+k}\binom{2L+k}{L}}$.