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I think, a sensical 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 (based on the closed-form-formula for the direct computation) 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 of the factorials at negative parameters.

For instance, we get $$ \small \begin{array}{r|lllll|lll} \text{index k} & \\ \hline\\ &&&& \text{ extension to negative indexes} \\ \cdots &\cdots \\ -4& 1 & 3+\frac 1{16} & 6+\frac 3{16}+\frac 1{81}& 10+\frac 6{16}+\frac 3{81}+\frac 1{256}&\ldots &\tiny \sum \overset ?= & -4! \\ -3& 1 & 2+\frac 18 & 3+\frac 28+\frac 1{27}& 4+\frac 38+\frac 2{27}+\frac 1{64}&\ldots &\tiny \sum \overset ?= & -3! \\ -2& 1 & 1+\frac 14& 1+\frac 14+\frac 19& 1+\frac 14+\frac 19 + \frac 1{16}& \ldots &\tiny \sum \overset ?= & -2! \\ -1& 1 & 0+\frac 12 & 0+0+\frac 13 & 0+0+0+\frac 14 & \ldots &\tiny \sum \overset ?= & -1! \\ \hline \\ &&&& \text{ triangle of Eulerian numbers} \\ 0& 1 & . & . & . &. &\tiny\sum = & 0! \\ 1& 1 & . & . & . &. &\tiny\sum = & 1! \\ 2& 1 & 1 & . & . &. &\tiny\sum = & 2! \\ 3& 1 & 4 & 1 & . &. &\tiny\sum = & 3! \\ 4& 1 & 11 & 11 & 1 &. &\tiny\sum = & 4! \\ \cdots & \cdots \\ \end{array} $$ and so on.

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*I have a more involved discussion in a hobby-treatize about the Eulerian-triangle [here][3]*

I think, a sensical 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 (based on the closed-form-formula for the direct computation) 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 of the factorials at negative parameters.

For instance, we get $$ \small \begin{array}{r|lllll|lll} \text{index $k$} & \\ \hline\\ &&&& \text{ extension to negative indexes} \\ \cdots &\cdots \\ -4& 1 & 3+\frac 1{16} & 6+\frac 3{16}+\frac 1{81}& 10+\frac 6{16}+\frac 3{81}+\frac 1{256}&\ldots &\tiny \sum \overset ?= & -4! \\ -3& 1 & 2+\frac 18 & 3+\frac 28+\frac 1{27}& 4+\frac 38+\frac 2{27}+\frac 1{64}&\ldots &\tiny \sum \overset ?= & -3! \\ -2& 1 & 1+\frac 14& 1+\frac 14+\frac 19& 1+\frac 14+\frac 19 + \frac 1{16}& \ldots &\tiny \sum \overset ?= & -2! \\ -1& 1 & 0+\frac 12 & 0+0+\frac 13 & 0+0+0+\frac 14 & \ldots &\tiny \sum \overset ?= & -1! \\ \hline \\ &&&& \text{ triangle of Eulerian numbers} \\ 0& 1 & . & . & . &. &\tiny\sum = & 0! \\ 1& 1 & . & . & . &. &\tiny\sum = & 1! \\ 2& 1 & 1 & . & . &. &\tiny\sum = & 2! \\ 3& 1 & 4 & 1 & . &. &\tiny\sum = & 3! \\ 4& 1 & 11 & 11 & 1 &. &\tiny\sum = & 4! \\ \cdots & \cdots \\ \end{array} $$ and so on.

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I have a more involved discussion in a hobby-treatise about the Eulerian-triangle here.

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 (based on the closed-form-formula for the direct computation) 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 of the factorials at negative parameters.

For instance, we get $$ \small \begin{array}{r|lllll|lll} \text{index k} & \\ \hline\\ &&&& \text{ extension to negative indexes} \\ \cdots &\cdots \\ -4& 1 & 2+\frac 1{16} & 6+\frac 3{16}+\frac 1{81}& 10+\frac 6{16}+\frac 3{81}+\frac 1{256}&\ldots &\tiny \sum \overset ?= & -3! \\ -3& 1 & 2+\frac 18 & 3+\frac 28+\frac 1{27}& 4+\frac 38+\frac 2{27}+\frac 1{64}&\ldots &\tiny \sum \overset ?= & -3! \\ -2& 1 & 1+\frac 14& 1+\frac 14+\frac 19& 1+\frac 14+\frac 19 + \frac 1{16}& \ldots &\tiny \sum \overset ?= & -2! \\ -1& 1 & 0+\frac 12 & 0+0+\frac 13 & 0+0+0+\frac 14 & \ldots &\tiny \sum \overset ?= & -1! \\ \hline \\ &&&& \text{ triangle of Eulerian numbers} \\ 0& 1 & . & . & . &. &\tiny\sum = & 0! \\ 1& 1 & . & . & . &. &\tiny\sum = & 1! \\ 2& 1 & 1 & . & . &. &\tiny\sum = & 2! \\ 3& 1 & 4 & 1 & . &. &\tiny\sum = & 3! \\ 4& 1 & 11 & 11 & 1 &. &\tiny\sum = & 4! \\ \cdots & \cdots \\ \end{array} $$ and so on.

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*I have a more involved discussion in a hobby-treatize about the Eulerian-triangle [here][3]*

I think, a sensical 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 (based on the closed-form-formula for the direct computation) 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 of the factorials at negative parameters.

For instance, we get $$ \small \begin{array}{r|lllll|lll} \text{index k} & \\ \hline\\ &&&& \text{ extension to negative indexes} \\ \cdots &\cdots \\ -4& 1 & 3+\frac 1{16} & 6+\frac 3{16}+\frac 1{81}& 10+\frac 6{16}+\frac 3{81}+\frac 1{256}&\ldots &\tiny \sum \overset ?= & -4! \\ -3& 1 & 2+\frac 18 & 3+\frac 28+\frac 1{27}& 4+\frac 38+\frac 2{27}+\frac 1{64}&\ldots &\tiny \sum \overset ?= & -3! \\ -2& 1 & 1+\frac 14& 1+\frac 14+\frac 19& 1+\frac 14+\frac 19 + \frac 1{16}& \ldots &\tiny \sum \overset ?= & -2! \\ -1& 1 & 0+\frac 12 & 0+0+\frac 13 & 0+0+0+\frac 14 & \ldots &\tiny \sum \overset ?= & -1! \\ \hline \\ &&&& \text{ triangle of Eulerian numbers} \\ 0& 1 & . & . & . &. &\tiny\sum = & 0! \\ 1& 1 & . & . & . &. &\tiny\sum = & 1! \\ 2& 1 & 1 & . & . &. &\tiny\sum = & 2! \\ 3& 1 & 4 & 1 & . &. &\tiny\sum = & 3! \\ 4& 1 & 11 & 11 & 1 &. &\tiny\sum = & 4! \\ \cdots & \cdots \\ \end{array} $$ and so on.

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*I have a more involved discussion in a hobby-treatize about the Eulerian-triangle [here][3]*

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

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 (based on the closed-form-formula for the direct computation) 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 of the factorials at negative parameters.

For instance, we get $$ \small \begin{array}{r|lllll|lll} \text{index k} & \\ \hline\\ &&&& \text{ extension to negative indexes} \\ \cdots &\cdots \\ -4& 1 & 2+\frac 1{16} & 6+\frac 3{16}+\frac 1{81}& 10+\frac 6{16}+\frac 3{81}+\frac 1{256}&\ldots &\tiny \sum \overset ?= & -3! \\ -3& 1 & 2+\frac 18 & 3+\frac 28+\frac 1{27}& 4+\frac 38+\frac 2{27}+\frac 1{64}&\ldots &\tiny \sum \overset ?= & -3! \\ -2& 1 & 1+\frac 14& 1+\frac 14+\frac 19& 1+\frac 14+\frac 19 + \frac 1{16}& \ldots &\tiny \sum \overset ?= & -2! \\ -1& 1 & 0+\frac 12 & 0+0+\frac 13 & 0+0+0+\frac 14 & \ldots &\tiny \sum \overset ?= & -1! \\ \hline \\ &&&& \text{ triangle of Eulerian numbers} \\ 0& 1 & . & . & . &. &\tiny\sum = & 0! \\ 1& 1 & . & . & . &. &\tiny\sum = & 1! \\ 2& 1 & 1 & . & . &. &\tiny\sum = & 2! \\ 3& 1 & 4 & 1 & . &. &\tiny\sum = & 3! \\ 4& 1 & 11 & 11 & 1 &. &\tiny\sum = & 4! \\ \cdots & \cdots \\ \end{array} $$ and so on.

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*I have a more involved discussion in a hobby-treatize about the Eulerian-triangle [here][3]*