A second approach gave the numbers in a completely surprising context; I've no explanation so far.
For the function $\exp(x)-1$ the matrix $S_2$ of Stirling numbers 2'nd kind is involved. I've just recently begun to exercise with the Jordan-decomposition, and for some exercise I took this matrix

$$ \large S_2 \qquad = \qquad \small{ \begin{bmatrix}
1 & . & . & . & . \\
1 & 1 & . & . & . \\
1 & 3 & 1 & . & . \\
1 & 7 & 6 & 1 & . \\
1 & 15 & 25 & 10 & 1
\end{bmatrix}} $$
and let WolframAlpha Jordan-decompose it such that $$S_2 = S \cdot J \cdot S^{-1}$$

This gave the three matrices:

$$ \begin{array} {} S &=&\small { \begin{bmatrix}
1 & . & . & . & . \\
0 & 1 & . & . & . \\
0 & 1 & 3 & . & . \\
0 & 1 & 13 & 18 & . \\
0 & 1 & 50 & 205 & 180
\end{bmatrix} } \\
J&=& \small {\begin{bmatrix}
1 & . & . & . & . \\
1 & 1 & . & . & . \\
0 & 1 & 1 & . & . \\
0 & 0 & 1 & 1 & . \\
0 & 0 & 0 & 1 & 1
\end{bmatrix} } \\ S^{-1}&=&\small {\begin{bmatrix}
1 & . & . & . & . \\
0 & 1 & . & . & . \\
0 & -1/3 & 1/3 & . & . \\
0 & 5/27 & -13/54 & 1/18 & . \\
0 & -301/2430 & 353/1944 & -41/648 & 1/180
\end{bmatrix}}\end{array}$$
(Note: I transposed the input to Woframalpha and then also the output to keep in line with my usual conventions with that type of matrix-discussion)

The surprise is: *that we find the coefficients in the matrix $S$*

For the use for the function $f(x) = \exp(x)-1$ the matrix $S_2$ is factorially similarity-scaled and becomes then the Bell/(transposed) Carleman-matrix for that function. In that view $S_2$ is a matrixoperator; and the matrices $S$ and $S^{-1}$ are in a very similar role as the operators for the Schröder-function of $f(x)$ which is indeed at the heart of the Schröder/Abel-type of fractional iteration...

Disclaimer: this is *only a hint*; I've not yet analyzed the actual rôle in such a setting of operators.

This is the input for W/A's input field:

JordanForm({{1,1,1,1,1},{0,1,3,7,15},{0,0,1,6,25},{0,0,0,1,10},{0,0,0,0,1}})

Mathematicafunction at OEIS works as desired. – Fred Kline Jun 13 '13 at 8:36