I was considering a modification of this structure to read:

If we call this a MacLaurin Generating Function, then we can get some interesting series at x=1...

It continues after 0,1,2,3 with 8,10,54... and I've just learned that these are the Lehmer-Comtet numbers: Lehmer-Comtet

Now we can see what other of these types of series might have simple MacLaurin generating functions, below is if you use x^(2*x)

Maybe not as complicated as it looks when you substitute in x=1 that last 7th derivative is:

Still pretty complicated but the non power of 2 numbers look to be related to multiples of factorials?

The series 1, 2, 6, 18, 64, 220, 888 is not in the Encyclopedia of Integer sequences, but looks interesting! In fact Alois P. Heinz has already pointed out that the exponential generating function for x^2*x is x^2*x itself!

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