A012393 E.g.f. arctanh(tan(x)*tan(x)) (even powers only).

0, 2, 16, 512, 34816, 4063232, 724566016, 183240753152, 62382319599616, 27507470234550272, 15250953398036463616, 10384044045105304174592, 8517992937742473694806016, 8285310016381680852100186112

Offset

0,2

Comments

a(n) = 2*A000828(2*n-1). - corrected by Vaclav Kotesovec, Feb 08 2015

Links

Table of n, a(n) for n=0..13.

Formula

E.g.f.: log(sec(2*x))/2, cf. A000182. - Vladeta Jovovic, Jun 04 2005

a(n) = (-1)^(n(2n+1))4^(2n+1)(E{2n+1}(1/2)+E{2n+1}(1)); E_{n}(x) Euler polynomial. - Peter Luschny, Nov 25 2010

G.f.: 2*x/G(0) where G(k) = 1 - (2*k+2)*(2*k+4)*x/G(k+1)) (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 25 2012

G.f.: (2/G(0) - 1)*sqrt(-x), where G(k) = 1 + 1/(1 - 1/(1 + 1/(4*sqrt(-x)*(k+1)) - 1/G(k+1))) (continued fraction). - Sergei N. Gladkovskii, May 29 2013

G.f.: 2*x*T(0), where T(k) = 1 - x*(2*k+2)*(2*k+4)/(x*(2*k+2)*(2*k+4) - 1/T(k+1) ) (continued fraction). - Sergei N. Gladkovskii, Oct 12 2013

a(n) ~ 2^(4*n) * (2*n-1)! / Pi^(2*n). - Vaclav Kotesovec, Feb 08 2015

Example

arctanh(tan(x)*tan(x)) = (2/2!)*x^2 + (16/4!)*x^4 + (512/6!)*x^6 + (34816/8!)*x^8 + ...

Maple

A012393 := n -> (-1)^(n*(2*n+1))*4^(2*n+1)*(euler(2*n+1, 1/2) + euler(2*n+1, 1)) end; # Peter Luschny, Nov 25 2010

Mathematica

nn = 20; Table[(CoefficientList[Series[ArcTanh[Tan[x]^2], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* Vaclav Kotesovec, Feb 08 2015 *)

Crossrefs

Sequence in context: A140311 A012389 A009710 * A189899 A189335 A189773

Adjacent sequences: A012390 A012391 A012392 * A012394 A012395 A012396

Keyword

nonn

Author

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

Extensions

a(0)=0 prepended by Joerg Arndt, Oct 26 2012

Status

approved