Exponential Generating Functions

[kensaurus]

I got a question here, and I am stuck at understanding the step of the solution. Any help will be appreciated.

http://img841.imageshack.us/img841/6589/40155869.jpg

I would like to know how to get from the second to the third step, where the summation comes in.

It looks like multiplication of 2 exponential generating function, but the second step is not one. Thanks for any help.

 

[eczeno]

the second step is the binomial expansion of (e^x-1)^k.

cheers

 

[kensaurus]

oh my, I am slapping my head... i went to derive it through complex, exponential functions and other means...

thanks alot!

 

[eczeno]

cheers.