Number of Distinct Resistances that can be produced from n equal resistance resisters

Here is an interesting problem:

The number of distince resistances that can be produced from n equal resistance resisters is given below.

The Sequence

Surprisingly this is also equal to the number of terms in nth order differentiation of a function y(x) with respect to x. How do I connect that these facts? That these represent the same counting and are expected to be equal.

P.S. I have only checked a few first terms of the differentiation. It may not be correct.

Edited. differentiation of a $nth$ power of function $y(x)$ with respect to $x$. E.G. $$\frac{dy^n}{dx}=ny^{n-1}y'$$

$$({y^n})''=n(n-1)y^{n-2}y'+ny^{n-1}y''$$ $$\vdots$$

ADDED Ok, if this sequence does not necessarily, explain this, how can I determine number of terms in this iteration? I found first five terms, if not mistaken to be $1, 2, 4, 9, 22, ...$

Added Serious! Error detected! Question is incorrect. Sorry!

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What kind of function y(x) are you referring to here, and how many terms have you checked? – Foo Barrigno May 6 '13 at 12:57
up to 5 terms. I can't imagine taking 6th one. – user45099 May 6 '13 at 13:06

You say you've checked the first $5$ terms. Note that an OEIS search yields a full $36$ sequences containing these $5$ terms, many of which begin with those $5$ terms. You could be asking this question about any one of those sequences, with equal (lack of) justification. Unless you have a specific reason to believe that these two phenomena are related, there's nothing to explain here.