appearance of strange coefficients from taking successive derivatives Taking successive derivatives of $\varphi(x)=e^{\frac{1}{\log(x)}}$ and looking at the derivatives expanded out term by term yields coefficients that I cannot find a pattern to.

What's the pattern? Why do these numbers appear?

$\varphi'(x) \implies 1 $
$ \varphi''(x) \implies 1,2,1$
$\varphi^{(3)}(x)\implies 1,6,9,6,2 $
$ \varphi^{(4)}(x) \implies 1,12,42,60,47,22,6$
$ \varphi^{(5)}(x) \implies1,20,130,360,515,450,260,100,24. $
 A: I'll start by writing the numbers out as
\begin{equation*}
\begin{array}{c}
&&&& 1 \\
&&& 1 & 2 & 1 \\
&& 1 & 6 & 9 & 6 & 2 \\
& 1 & 12 & 42 & 60 & 47 & 22 & 6 \\
1 & 20 & 130 & 360 & 515 & 450 & 260 & 100 & 24 \\
&&&&\dots
\end{array}
\end{equation*}
and calculating, as suggested in the comments,
\begin{equation*}
 \frac{\mathrm d}{\mathrm dx} \left[\frac{\exp(1 / \log x)}{x^a (\log x)^b}\right]
= -\left(
  \frac{a \cdot \exp(1 / \log x)}{x^{a + 1} (\log x)^b} +
  \frac{b \cdot \exp(1 / \log x)}{x^{a + 1} (\log x)^{b + 1}} +
  \frac{\exp(1 / \log x)}{x^{a + 1} (\log x)^{b + 2}}
  \right)
\end{equation*}
So with the way I've presented the coefficients, we can find a "Pascal's Triangle"-like rule, where each number is determined by the three above it.
To be exact, if $C_{m, n}$ is the $m$th coefficient in the $n$th row (which is the coefficient of $\exp(1 / \log x) / (x^n (\log x)^{2n - m + 1})$, it's determined by
\begin{equation*}
 C_{m,n} = (n - 1)C_{m - 2, n - 1} + (2n - m)C_{m - 1, n - 1} + C_{m, n - 1}
\end{equation*}
where if an index goes "out of range" we treat it as $0$, which makes sense given the interpretation as a coefficient that does not appear.
An example of this recurrence is
\begin{equation*}
 360 = (5 - 1) \cdot 12 + (2 \cdot 5 - 4) \cdot 42 + 60
\end{equation*}
or
\begin{equation*}
 515 = (5 - 1) \cdot 42 + (2 \cdot 5 - 5) \cdot 60 + 47
\end{equation*}
