My question was intended somewhat along the line: Assume the Gamma function is not yet invented and Goldbach asks you the question: "What is (-n)! ?" You know that Goldbach expects a combinatorial answer in the domain of integer or rational numbers. What would you answer? I will give my answer in this sens.

Looking at GKP's ConMath, Table 253, the combined Stirling triangles in their dual form, we see: If we sum the columns in this triangle for k < 0 we get the factorial numbers, if we sum the rows for k > 0 we get the Bell numbers.

What about saying the Bell numbers are the factorial numbers at negative integers? Is the answer encoded in one of the most important triangles in combinatorics?

See what Knuth says about the origin of this duality (table on page 11).

```
{120}
. {24}
1, . {6}
10, 1, . {2}
35, 6, 1, . {1}
50, 11, 3, 1, . {1}
24, 6, 2, 1, 1, .
0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 1,.....{1}
0, 0, 0, 0, 0, 0, 1, 1,....{2}
0, 0, 0, 0, 0, 0, 1, 3, 1,....{5}
0, 0, 0, 0, 0, 0, 1, 7, 6, 1,....{15}
0, 0, 0, 0, 0, 0, 1, 15, 25, 10, 1,....{52}
```

`>`

) for emphasis. Quotes are for quoting. $\endgroup$ – Federico Poloni Mar 6 '18 at 8:47