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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.

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# How to restore the original formula from a binomial-like expansion?

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.