A034428 E.g.f.: 1 - (1-x)*(tan(x) + sec(x)).

0, 0, 1, 1, 3, 9, 35, 155, 791, 4529, 28839, 201939, 1542739, 12767689, 113794603, 1086657403, 11068604847, 119790363489, 1372696498127, 16603828720547, 211406514019115, 2826296899863929, 39584082775592211, 579600224535319371

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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

References

• R. Ehrenborg and S. Mahajan, Maximizing the descent statistic, Annals Combin., 2 (1998), no. 2, 111-129.

Links

• T. D. Noe, Table of n, a(n) for n = 0..102
• Miklós Bóna and István Mező, Limiting Probabilities for Vertices of a Given Rank in 1-2 Trees, The Electronic Journal of Combinatorics (2019) Vol. 26, No. 3, P#3.41.
• Peter J. Cameron and Liam Stott, Trees and cycles, arXiv:2010.14902 [math.CO], 2020. See p. 25.
• Richard Ehrenborg and Swapneel Mahajan, Maximizing the descent statistic, preprint.
• Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
• Zhicong Lin, Shi-Mei Ma, David G. L. Wang, and Liuquan Wang, Positivity and divisibility of alternating descent polynomials, arXiv:2011.02685 [math.CO], 2020.
• Ran Pan and Jeffrey Remmel, Counting alternating permutations with restricted prefix and suffix, arXiv:2502.10727 [math.CO], 2025. See p. 3.

Crossrefs

Essentially the same as A131281(n)/2.

Programs

Mathematica

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 *)

PARI

a(n)=n!*polcoeff(1-(1-x)*(tan(x+x*O(x^n))+1/cos(x+x*O(x^n))),n)

Formula

E.g.f.: 1 - (1-x)*(tan(x) + sec(x)).

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

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

a(n) ~ n!*(2 - 4/Pi)*(2/Pi)^n. - Vaclav Kotesovec, Jun 01 2013