I think, a better definition stems from the generalization of the triangle of eulerian numbers. For positive integer indexes the rows sum to factorials, and even if the rows are interpolated to fractional indexes the rowsums are fractional factorials or gamma values. Thus I assume the extension of the eulerian triangle to negative indexes gives the answer to a sensical definition if the factorials at negative parameters. For instance, we get -1! = 1 + 1/2 + 1/3 + 1/4 + ... -2! = (1) + (1+1/4) + (1+1/4+1/9) + ... -3! = (1) + (2+1/8) + (3+2/8+1/27) + ... and so on. The terms can be computed by the direct definition of Eulerian-numbers (see formula in wikipedia,for instance). I have discussed this in a hobby-treatize of the Eulerian-triangle in http://go.helms-net.de/math/binomial_new/01_12_Eulermatrix.pdf
Gottfried Helms