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Is there any closed form for :

$$\sum_{k=0}^{n}\binom{\binom{n}{k}}{k}$$

I have no idea how to start and I used Wolframalpha, but did not get any answer, any help is appreciated (it would be very nice if there exist a closed form just by using binomial identities).

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Check Petkovsek, Wilf, Zeilberger, "A= B", it gives algorithms that give the solution to sums of hypergeometric terms or prove they can't be written in closed form. Most computer algebra systems have packages for it, e.g. it is in Maxima (the computation involves checking potentially hundreds of possible keys, not for hand checking). Would need to check it is/massage into the right form, however.

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