Consider the following functions:
$f_0(x)=1$
$f_1(x)=\sum_{i=2}^x [(i-1) \cdot f_0(i)] = \frac{(x-1)x}{2}$
$f_2(x)=\sum_{i=2}^x [(i-1) \cdot f_1(i)] = \sum_{i=2}^x \frac{(i-1)^2i}{2} = \frac{(x-1)x(x+1)(3x-2)}{24}$
And so on.
Essentially: $f_{k}(x)=\sum_{i=2}^x [(i-1) \cdot f_{k-1}(i)]$ and $f_0(x)=1$ as the base case, where $x$ and $k$ are natural numbers and $x\ge2$.
I am trying to find a closed form formula for $f_k(x)$ that only depends on $k$ and $x$, but I'm not sure if this is possible at all. Is there a name for this recurrence or is there anything similar I could read on that could be helpful in solving this, I've tried all I could think of, so any tips or advice in the right direction would be great.
The ordinary generating function of the sequence $f_k(x)$ is defined as: $$ F(x,t) := \sum_{k=0}^\infty f_k(x) t^k = (-t)^{-x} \Gamma(-1/t)/\Gamma(x-1/t).$$ For example, $$F(0,t) = F(1,t) = 1,\; F(2,t) = (1-t)^{-1} = t^{-2}\Gamma(-1/t)/\Gamma(2-1/t),$$ $$F(3,t) = (1-t)^{-1}(1-2t)^{-1}\!,\; F(4,t)=(1-t)^{-1}(1-2t)^{-1}(1-3t)^{-1}\!. $$
The coefficients of $\;1/F(x,t)\;$ involve the Stirling numbers of the first kind, and of $\;F(x,t)\;$ the Stirling numbers of the second kind. For example, the coefficients of $\;F(4,t)\;$ is OEIS sequence A000392 (with a different offset). This implies that $\;f_k(4) = S(k+3,3),\;$ and in general we get $\;f_k(n+1) = S(k+n,n)\;$ where $S$ is Stirling numbers of the second kind.