I encountered with a recursive formula of the following kind:

$$A(0,x)=1$$

$$A(n,x)= \sum _{j=0}^{n-1} \binom{n-1}{j} A(n-j-1,x) \sum _{k=0}^{x-1} A(j,k)$$

The sum terms can be re-arranged so to get the following expression under the external sum:

$$\sum _{j=0}^{n-1}\binom{n-1}{j} A(n-j-1,x)A(j,k)$$

Which seems to be perfectly similar to the formula for a power of a binomial or for a derivative of a product with an exception that $A(n,x)$ is neither power nor derivative.

But one can suppose that this is in fact an expansion of a form

$$A(n-1,x*k)$$

where $*$ is some binary operation and $A$ plays the role of power.

So my question is whether it possible to recover this expression from the binomial-like expansion and if it possible please give me some hints on a research or summary about common name and properties of binomial-like expansions, which properties should have two operations so that their combination gives binomial-like formula etc.

An approach to find a general formula for $A(n,x)$ using generating functions apparently leads to nothing useful.