I am looking for a function $f_m(x)$ described by the following power series. Its coefficients look similar to the Bell polynomials; however, my attempts to rewrite it using the Bell polynomials or the multinomial theorem did not produce meaningful results.
\begin{equation} f_m(x) = \sum_{p=1}^{\infty} \underbrace{\frac{\sum \prod_{i=1}^p \frac{1}{k_i!}}{p+1}}_{:= a_{p,m}} x^p \end{equation}
The sum in the numerator of the fraction is taken over all sequences of non-negative integers $k_1, \dots, k_p$ which fulfill
\begin{align} p &= \sum_{i=1}^p i \cdot k_i \\ m &= \sum_{i=1}^p k_i \qquad m \in \mathbb{N}, m \geq 1 \end{align}
My question: Is $f_m(x)$ an already well-known and studied function or a composition of such functions?
EDIT
I just found that the coefficient of the power series, $a_{p,m}$, is zero for $m > p$. In addition, it appears that the numerator can be rewritten as \begin{align} \sum \prod_{i=1}^p \frac{1}{k_i!} = \frac{(p-1)!}{(p-m)!(m-1)!m!} \end{align}
Hence, $f_m(x)$ can be restated: \begin{align} f_m(x) &= \sum_{p=m}^{\infty} \frac{(p-1)!}{(p-m)!(m-1)!m!(p+1)} x^p\\ &= x^m \sum_{p=0}^{\infty} \frac{(p+m-1)!}{p!(m-1)!m!(p+m+1)} x^p\\ &= \frac{1}{x} \frac{x^{m+1}}{m!} \sum_{p=0}^{\infty} \binom{p+m-1}{m-1}\frac{1}{p+m+1} x^p\\ &= \frac{1}{x} \int dx \,\frac{x^{m}}{m!} \underbrace{\sum_{p=0}^{\infty} \binom{p+m-1}{m-1} x^p}_{= (1-x)^{-m}}\\ &= \frac{1}{x} \int dx \,\frac{1}{m!} \left(\frac{x}{1-x}\right)^m\\ &= \frac{x^m}{(m+1)!} \, {_2}F_1(m,m+1;m+2,x) \end{align} Thanks to TravorLZH for pointing out the steps in the last two lines.