Let $$$C_i$$$ be $$$i^{th}$$$ catalan number. Is it possible to derive a generalised formula for convolution of catalan numbers in $$$k$$$ variables.
For example:

For $$$K = 2$$$,
$$$\sum_{i=0}^{n} {C_iC_{n-i}} = C_{n+1}$$$

For $$$K = 3$$$,
$$$\sum_{i=0}^{n} \sum_{j=0}^{n} {C_iC_jC_{n-i-j}} = C_{n+2} - C_{n+1}$$$

For $$$K = 4$$$,
$$$\sum_{i=0}^{n} \sum_{j=0}^{n} \sum_{k=0}^{n} {C_iC_jC_kC_{n-i-j-k}} = C_{n+3} - {2 C_{n+2}}$$$

I wanted to know, does any formula exist for any generalized K?