most recent 30 from http://mathoverflow.net 2013-05-19T13:18:24Z http://mathoverflow.net/feeds/question/57615 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57615/about-power-series-for-iterated-logarithms Mark Sapir 2011-03-06T23:24:05Z 2011-03-07T23:51:56Z

<p>The question is motivated by <a href="http://mathoverflow.net/questions/57588/series-for-loglog-closed" rel="nofollow"> this one. </a> 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 <a href="http://oeis.org/search?q=1%2C-2%2C7%2C-35%2C228%2C+1834&amp;language=english&amp;go=Search" rel="nofollow">Sloane's encyclopedia </a> (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:</p> <p>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. </p>

http://mathoverflow.net/questions/57615/about-power-series-for-iterated-logarithms/57618#57618 Gerry Myerson 2011-03-06T23:55:49Z 2011-03-06T23:55:49Z

<p>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 <a href="http://oeis.org/A000268" rel="nofollow">http://oeis.org/A000268</a> 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). </p>