Yes, but the function ExponentialGeneratingFunction does not compute the exponential generating function from a set of initial terms.

@LLIAMnYP: Oh, I see, you have created the function findExponentialGeneratingFunction.

Thanks a lot to you all for your answers and comments. In fact, I had tried FindGeneratingFunction with the terms divided by the corresponding factorials, but it did not work. Apparently, it needed more terms. That was the reason for my second question. As for the function "findExponentialGeneratingFunction", where is it? I can't find it. Please excuse my ignorance :-)