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The question is motivated by this one. It turned out (see my comment there) that the coefficients of the Taylor series for $\log\log x$ at $x=e$ have nice combinatorial description from Sloane's encyclopedia (in the encyclopedia, a related but slightly more complicated function is considered). The coefficients are (up to a power of $e$ multiplied by a factorial) permanents of some easily defined matrices. My question is this:

Is there a combinatorial (possibly 3-dimensional) description of coefficients of the Taylor series of $\log\log\log x$ at $e^e$? Same question for $\log\log\log\log x$, etc.

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Couldn't edit to correct spelling - it's "Sloane's encyclopedia" (N.J.A.Sloane) –  Gottfried Helms Mar 7 '11 at 8:55
    
@Gottfried: OK, thanks. I fixed that. –  Mark Sapir Mar 7 '11 at 23:53
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up vote 3 down vote accepted

I think it may be simpler to deal with the Maclaurin series for the functions $\log(1-x)$, $-\log(1+\log(1-x))$, $-\log(1+\log(1+\log(1-x)))$, etc. The third one, for example, is the exponential generating function for $1,3,15,105,947,10472,137337,\dots$ which is http://oeis.org/A000268 and there are a couple of references there which may be worth tracking down, J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353 and P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5 (and a couple of others, besides).

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Thanks! It seems to be related to A039815 which is a 3-dimensional object. Still it is not a "permanent" of any 3-dimensional matrix yet. Perhaps there exists such a thing. –  Mark Sapir Mar 7 '11 at 0:02
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