I stumbled upon a problem that can be distilled to:
Let $\Delta_m(x)$ be some function that depends on $x$ such that
$$ \delta_{x}\Delta_m(x) = \Delta_{m+1}(x)$$
where $\delta_{x} $ is the derivative with respect to $x$. I'm interested in solving exactly
$$(x\delta_{x})^m[\Delta_0(x)]$$
where $(x\delta_{x})^m$ denotes $m$ applications of the operator $(x\delta_{x})$ onto $\Delta_0(x)$.
Doing a few applications by hand yields (I omit the $x$ dependency for clarity)
$$ x\Delta_1$$ $$ x\Delta_1+x^2\Delta_2$$ $$ x\Delta_1+x^2\Delta_2+2x^2\Delta_2 +x^3\Delta_3$$ etc...
Arranging coefficients in a triangle by powers of $x^n \Delta_n$ yields a "Pascal-triangle-like" diagram
$$ 1 $$ $$ 1\ 1\ $$ $$ 1\ 3\ 1\ $$ $$ 1\ 7\ 6\ 1$$ $$ 1\ 9\ 25\ 10\ 1$$ $$ ... $$
for which I cannot seem to find the combinatorial expression. What would be the general expression for the $m^{th}$ row?