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%I #22 Jul 28 2018 10:38:09
%S 1,1,3,13,67,421,3115,26349,250867,2655541,30929019,393019837,
%T 5410699075,80221867909,1274393162827,21594697199757,388796268801427,
%U 7411769447027413,149143210226032923,3159088788867736669
%N Expansion of e.g.f. exp(x) * (sec(exp(x) - 1))^2.
%C Take the smallest element from each block of the set partitions of {1,2,...,n+1} into an odd number of blocks. Form a "zag" 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 such "zag" permutations. - _Geoffrey Critzer_, Nov 23 2012
%H Muniru A Asiru, <a href="/A080832/b080832.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 E.g.f.: exp(x) / (cos(exp(x) - 1))^2.
%F The sequence 0, 1, 1, 3, ... has e.g.f. tan(exp(x)-1). It has general term sum{k=0..n, S2(n, k) A009006(k)} for n>1 (S2(n, k) Stirling numbers of second kind). - _Paul Barry_, Apr 20 2005
%F a(n) ~ 2*n * n! / ((2+Pi) * (log(1+Pi/2))^(n+2)). - _Vaclav Kotesovec_, Jul 28 2018
%p seq(coeff(series(factorial(n)*exp(x)*(sec(exp(x)-1))^2, x,n+1),x,n),n=0..25); # _Muniru A Asiru_, Jul 28 2018
%t nn=21;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Drop[Range[0,nn]!CoefficientList[ Series[Tan[Exp[x]-1],{x,0,nn}],x],1] (* _Geoffrey Critzer_, Nov 23 2012 *)
%Y Cf. A000182, A219613.
%K easy,nonn
%O 0,3
%A _Emanuele Munarini_, Mar 28 2003