A012299 - OEIS

%I #20 Mar 28 2024 21:54:08

%S 0,2,-8,-88,6592,-9568,-49063808,4426189952,1122968737792,

%T -441081682390528,-23926396899780608,74405808039377364992,

%U -16597462789247237931008,-19016633437725878038847488

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

%H Vaclav Kotesovec, <a href="/A012299/b012299.txt">Table of n, a(n) for n = 0..220</a>

%H Vaclav Kotesovec, <a href="/A012299/a012299.jpg">graph a(n) / asymptotic</a>.

%F 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

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

%t Table[n!*SeriesCoefficient[ArcSinh[Sin[x]*Sin[x]],{x,0,n}],{n,0,40,2}] (* _Vaclav Kotesovec_, Nov 02 2013 *)

%o (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

%o (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

%K sign

%O 0,2

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

%E Missing a(0)=0 prepended by _Vaclav Kotesovec_, Nov 02 2013