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%I #26 Dec 02 2013 17:30:51
%S 0,1,2,8,48,376,3600,40592,525952,7692928,125303040,2248366592,
%T 44055035904,935800603648,21417131939840,525346642337792,
%U 13748654428323840,382362034331877376,11260657076602208256,350082293087247269888,11457214800338786713600
%N E.g.f. tan(x/(1-x)).
%C Take each set partition of {1,2,...,n} into an odd number of blocks. Linearly order the elements within each block then form a "zag" permutation with the smallest element from each block. Here a "zag" permutation is a permutation a[1],a[2],...,a[k] such that a[1] < a[2] > a[3] < ... > a[k]. a(n) is the number of ways to order the blocks in accordance with each "zag" permutation.
%H Vincenzo Librandi, <a href="/A219613/b219613.txt">Table of n, a(n) for n = 0..100</a>
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 144
%F a(n) ~ 4/(Pi*(2+Pi))* n! * (1+2/Pi)^n. - _Vaclav Kotesovec_, Nov 25 2012
%F E.g.f.: x/(1-x)/T(0), where T(k) = 4*k+1 - x^2/((4*k+3)*(1-x)^2 - x^2/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 30 2013
%e a(3) = 8: The set partitions of {1,2,3} into an odd number of blocks are {1,2,3} and {1}{2}{3}. There are 6 ways to linearly order the elements of {1,2,3}. There are 2 such ways to order the blocks of the set partition {1}{2}{3}: {1}{3}{2} and {2}{3}{1}. 6+2=8.
%t nn=21;Range[0,nn]!CoefficientList[Series[Tan[x/(1-x)],{x,0,nn}],x]
%Y Cf. A000182, A080832.
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Nov 23 2012
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