about power series for iterated logarithms - MathOverflow 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 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 Answer by Gerry Myerson for about power series for iterated logarithms 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>