%I #62 Aug 06 2024 21:36:16
%S 0,0,1,1,3,9,35,155,791,4529,28839,201939,1542739,12767689,113794603,
%T 1086657403,11068604847,119790363489,1372696498127,16603828720547,
%U 211406514019115,2826296899863929,39584082775592211,579600224535319371
%N E.g.f.: 1 - (1-x)*(tan(x) + sec(x)).
%C Also: number of permutations on n elements having the descent pattern: up, up, down, up, down, ...; a(n) = n * E_{n-1} - E_{n} where E_{n} denotes the Euler numbers, see sequence A000111. - _Richard Ehrenborg_, Feb 12 2002
%D R. Ehrenborg and S. Mahajan, Maximizing the descent statistic, Annals Combin., 2 (1998), no. 2, 111-129.
%H T. D. Noe, <a href="/A034428/b034428.txt">Table of n, a(n) for n = 0..102</a>
%H Miklós Bóna and István Mező, <a href="https://doi.org/10.37236/7731">Limiting Probabilities for Vertices of a Given Rank in 1-2 Trees</a>, The Electronic Journal of Combinatorics (2019) Vol. 26, No. 3, P#3.41.
%H Peter J. Cameron and Liam Stott, <a href="https://arxiv.org/abs/2010.14902">Trees and cycles</a>, arXiv:2010.14902 [math.CO], 2020. See p. 25.
%H R. Ehrenborg and S. Mahajan, <a href="http://www.math.iitb.ac.in/~swapneel/Perm.pdf">Maximizing the descent statistic</a>, preprint.
%H M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
%H Zhicong Lin, Shi-Mei Ma, David G. L. Wang, and Liuquan Wang, <a href="https://arxiv.org/abs/2011.02685">Positivity and divisibility of alternating descent polynomials</a>, arXiv:2011.02685 [math.CO], 2020.
%F E.g.f.: 1 - (1-x)*(tan(x) + sec(x)).
%F E.g.f.: E(x) = x + x*(x-1)/U(0) where U(k) = 4k + 1 - x/(2 - x/(4k + 3 + x/(2 + x/U(k+1)))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 22 2012
%F E.g.f.: x + 2*x*(x-1)/(U(0)-x) where U(k) = 4*k+2 - x^2/U(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jan 31 2013
%F a(n) ~ n!*(2 - 4/Pi)*(2/Pi)^n. - _Vaclav Kotesovec_, Jun 01 2013
%t With[{nn=30},Drop[CoefficientList[Series[1-(1-x)(Tan[x]+Sec[x]),{x,0,nn}], x]Range[0,nn]!,2]] (* _Harvey P. Dale_, Jan 22 2012 *)
%o (PARI) a(n)=n!*polcoeff(1-(1-x)*(tan(x+x*O(x^n))+1/cos(x+x*O(x^n))),n)
%Y Essentially the same as A131281(n)/2.
%K nonn,easy,nice
%O 0,5
%A _N. J. A. Sloane_