Algorithm to generate the number of tuples that satisfy the sum equals product problem.
One simple optimization you can use is as follows: since $x_1 + \cdots + x_n \leq 2n$, it follows that at most $\log_3 (2n)$ values are at least 3. You can go over all possible values for these $O(\log n)$ elements, and for each of them it should be easy to determine whether they can be completed (by adding only 1s and 2s) to a solution.

Taking this one step further, you can start adding elements to the sequence, from $x_n$ down, keeping track of the current product. If it exceeds $2n$, you can immediately halt.

You can probably optimize this much further, but this is already a good start.

Alternatively, OEIS has your sequence as A033178, and a table for $n \leq 10000$.