Combinatorics: Generating functions

[BrownianMan]

I have that B(x)=e^{e^{x}-1} is the generating function for the number of set partitions. Also the Stirling numbers of the second kind are defined by S(0,0)=1, S(n,0)=S(0,n)=0 for n=>1 and S(n,k)=S(n-1, k-1) + kS(n-1, k). Show that

e^{u(e^{x}-1)}=1+\sum_{n\geq 1}\sum_{k=1}^{n}S(n,k)u^{k}\frac{x^{n}}{n!}

Use this result to show that

S(n,k)=\frac{1}{k!}\sum_{r=0}^{k}(-1)^{k-r}\left \binom{k}{r}r^{n}

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I've tried doing this but I cannot seem to get very far. Could someone show me how to prove this? Thanks.

 

[haruspex]

So
e^{u(e^{x}-1)}=B(x)^u
If you write that out using the definition of B(x), collecting up the coefficients of xⁿ will, with luck, produce:
\frac{1}{n!}\sum_{k=1}^{n}S(n,k)u^{k}