A012299 Expansion of e.g.f. arcsinh(sin(x)*sin(x)), even-indexed terms only.

0, 2, -8, -88, 6592, -9568, -49063808, 4426189952, 1122968737792, -441081682390528, -23926396899780608, 74405808039377364992, -16597462789247237931008, -19016633437725878038847488

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..220

Vaclav Kotesovec, graph a(n) / asymptotic.

Formula

Lim sup n->oo (|a(n)|*n^(3/2)/(2*n)!)^(1/(2*n)) = 1.04762030856875... = 1/sqrt(arcsin(sqrt(1-1/sqrt(2)))^2 + (log(1+sqrt(2)-sqrt(2*(1+sqrt(2))))/2)^2). - Vaclav Kotesovec, Nov 02 2013

Example

E.g.f. = 2*x^2/2! - 8*x^4/4! - 88*x^6/6! + 6592x^8/8! + ...

Mathematica

Table[n!*SeriesCoefficient[ArcSinh[Sin[x]*Sin[x]], {x, 0, n}], {n, 0, 40, 2}] (* Vaclav Kotesovec, Nov 02 2013 *)

Prog

(PARI) x='x+O('x^50); v=Vec(serlaplace(asinh(sin(x)^2))); concat([0], vector(#v\2, n, v[2*n-1])) \\ G. C. Greubel, Oct 25 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Argsinh(Sin(x)^2) ));  [0] cat [Factorial(2*n+2)*b[2*n+1]: n in [0..Floor((m-4)/2)]];  // G. C. Greubel, Oct 25 2018

Crossrefs

Sequence in context: A136749 A226321 A054955 * A012295 A009486 A110384

Adjacent sequences: A012296 A012297 A012298 * A012300 A012301 A012302

Keyword

sign

Author

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

Extensions

Missing a(0)=0 prepended by Vaclav Kotesovec, Nov 02 2013

Status

approved