A012816 E.g.f. arctan(sec(x)*sinh(x)) (odd powers only).

1, 2, -20, -488, 22160, 1616672, -172976960, -25518205568, 4964227109120, 1231298393825792, -379260096755225600, -142026494757146421248, 63547531933929827962880, 33481297996129270926221312, -20517021964757071715832381440, -14468510293983989090015078678528

Offset: 0

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Keywords

• sign

Comments

The unsigned sequence {|a(n)|}n>=1 = [2,20,488,22160,...] enumerates binary increasing trees on 2*n vertices with a perfect matching (Kuba and Wagner).

Examples

			arctan(sec(x)*sinh(x)) = x+2/3!*x^3-20/5!*x^5-488/7!*x^7+22160/9!*x^9...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..228
• M. Kuba and S. Wagner, Perfect matchings and k-indecomposabilty of increasing trees, Seminaire Lotharingien de Combinatoire 57 (2007), Article B57a.

Crossrefs

Bisection (odd part) of A009342.

Programs

Maple

a:= n-> (2*n+1)! *
    coeff(series(arctan(sec(x)*sinh(x)), x, 2*(n+1)), x, 2*n+1):
seq(a(n), n=0..20);

Mathematica

With[{nn=40},Take[CoefficientList[Series[ArcTan[Sec[x]Sinh[x]],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* Harvey P. Dale, Jul 25 2024 *)

Formula

1/cosh(x*sqrt(2)) = 1 - 2x^2/2! + 20*x^4/4! - 488*x^6/6! +-...

a(n) = (-1)^[n/2]*2^n*A000364(n). - Philippe Deléham, Jun 16 2007

G.f. (for the unsigned sequence): 1/G(0) where G(k) = 1 - 2*x*(k+1)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 12 2013

G.f. (for the unsigned sequence): Q(0), where Q(k) = 1 - 2*x*(k+1)^2/(2*x*(k+1)^2 - 1/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 10 2013

E.g.f.(for the unsigned sequence, odd powers only): 1 + T(0)*x^2 /(1-x^2), where T(k) = 1 - 2*x^2*(2*k+1)*(2*k+2)/( 2*x^2*(2*k+1)*(2*k+2) + ((2*k+1)*(2*k+2)-2*x^2)*((2*k+3)*(2*k+4)-2*x^2)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 14 2013