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The sum $\sum_{j=0}^n \binom{n}{j} \left\{ j \atop k \right\} x^j$

A recent answer of mine to a question on Math Overflow includes the sum $$S(n,k,x) = \sum_{j=0}^n \binom{n}{j} \left\{ j \atop k \right\} x^j,$$ where $\left\{ j \atop k \right\}$ is a Stirling number ...
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An identity involving Stirling numbers of the second kind and binomial coefficients

Need to prove: $$\sum\limits_{k=0}^{n} \binom nk k^r x^k = \sum\limits_{j=0}^{r} \binom nj j! (1+x)^{n-j} x^j S(r,j)$$ where $S (n, k)$ denotes a Stirling number of second kind, the number of ...
Prove the following identity $$\displaystyle \sum_{i+j=m}\frac{(n-1) \binom{ai+n-1}{i} \binom{aj+1}{j}}{(ai+n-1)(aj+1)} = \frac{n\binom{am+n}{m}}{am+n}$$ where $i = 0,1,\cdots,m$ and $m, n$ are ...