I need to know if a function exists that partitions terms in such a way as seen below
$$ \frac{d^n}{dx^n}[\frac{(x)_c}{n!}] $$ Note that $(x)_c$ is the falling factorial of x and $c \geq n$, This in turn partitions the terms within the falling factorial. Here is an example:
$$ \frac{d^2}{dx^2}[\frac{(x)_4}{2!}] = \frac{d^2}{dx^2}[\frac{x(x-1)(x-2)(x-3)}{2}]$$
$$ =\frac{1}{2} \frac{d}{dx}[ x(x-1)(x-2)+x(x-2)(x-3)+x(x-1)(x-3)+(x-1)(x-2)(x-3)]$$
$$ = \frac{1}{2} [2x(x-1)+2x(x-2)+2x(x-3)+2(x-1)(x-2)+2(x-2)(x-3)+2(x-1)(x-3)]$$
$$ = x(x-1)+x(x-2)+x(x-3)+(x-1)(x-2)+(x-1)(x-2)+(x-2)(x-3)$$
This seems similar to bell polynomials or something of the sort, I don't know if such a function even exists. All help will be appreciated, Thank you.