Let $S(n, j)$ be Stirling's number of second kind. Let $p\in [0,1]$ and $m \in N$.
Bound from above the following sum:
$$
\sum\_{j=0}^m S(n,j) {m \choose j}\, j! \, p^j
$$



Estimation of a sum involving Stirling's number of second kind and binomial coefficient
*This is not yet a full answer, but more an expanded comment, asking firstly whether I've your formula correctly translated.*


If I get your formula right, then, for $p=1$ , the results written in form of a matrix-problem looks like
$$ X(1) = S2 \cdot \,^dF \cdot P = \small \begin{bmatrix} 
 1 & 1 & 1 & 1 & 1 & 1 & \cdots\\ 
 0 & 1 & 2 & 3 & 4 & 5 \\ 
 0 & 1 & 4 & 9 & 16 & 25 \\ 
 0 & 1 & 8 & 27 & 64 & 125 \\ 
 0 & 1 & 16 & 81 & 256 & 625 \\ 
 0 & 1 & 32 & 243 & 1024 & 3125 \\
 \vdots & & & & & & \ddots \end{bmatrix}
$$


where $S2$ is the matrix of Stirlingnumbers 2nd kind, $F=[0,1,2!,3!,...]$ the factorials and $\,^dF$ taken as diagonal, and $P$ the upper triangular Pascalmatrix.


If this is correct, then your problem is, with a vector $V(p)=[1,p,p^2,...]$ and its diagonal $\,^dV(p)$ we would ask for estimates in $X(p)$:
$$ X(p) = S2 \cdot \,^dF \cdot \,^dV(p) \cdot P $$
The occuring pattern in $X(p)$ ist not easily to decode, but we can rewrite the matrixformula a bit
$$\begin{array} {} X(p) &= S2 \cdot \,^dF \cdot \,^dV(p) \cdot P \\
&= S2 \cdot \,^dF \cdot (P \cdot P^{-1} \cdot) \,^dV(p) \cdot P \\
&= X(1) \cdot ( P^{-1} \cdot \,^dV(p) \cdot P )\\
&= X(1) \cdot P \cdot \,^dV(p) \\
\end{array}$$
and the constant matrix from the two beginning factors looks like
$$ X(1) \cdot P =\small \begin{bmatrix} 
 1 & 2 & 4 & 8 & 16 & 32 \\ 
 0 & 1 & 4 & 12 & 32 & 80 \\ 
 0 & 1 & 6 & 24 & 80 & 240 \\ 
 0 & 1 & 10 & 54 & 224 & 800 \\ 
 0 & 1 & 18 & 132 & 680 & 2880 \\ 
 0 & 1 & 34 & 342 & 2192 & 11000
 \end{bmatrix}
$$
where we only need to multiply the columns by the parameter $p^c$ where $c$ is the column-number starting from $0$ to get your sought values.  

I think a formula for the entries of $ X(1) \cdot P$ is not so difficult to derive and thus to have the upper bounds.


