A012271 - OEIS

%I #15 Sep 08 2022 08:44:38

%S 0,0,2,-6,22,-100,428,-1008,-12504,325152,-5267448,72020520,

%T -835748520,6577169040,36671947440,-3513587807520,114499586712000,

%U -2893590831936000,61563908486205600,-1035793335840588000

%N Expansion of e.g.f. arcsinh(log(x+1)*log(x+1)).

%H G. C. Greubel, <a href="/A012271/b012271.txt">Table of n, a(n) for n = 0..425</a>

%F Lim sup n->infinity (|a(n)|/n!)^(1/n) = 1.4236247666... = exp(1/sqrt(2))/sqrt(1+exp(sqrt(2)) - 2*exp(1/sqrt(2))*cos(1/sqrt(2))). - _Vaclav Kotesovec_, Nov 02 2013

%e E.g.f. = 2*x^2/2! - 6*x^3/3! + 22*x^4/4! - 100*x^5/5! + ...

%p seq(coeff(series(factorial(n)*arcsinh(log(x+1)*log(x+1)),x,n+1), x, n), n = 0 .. 20); # _Muniru A Asiru_, Oct 28 2018

%t CoefficientList[Series[ArcSinh[Log[x+1]*Log[x+1]], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 30 2013 *)

%o (PARI) x='x+O('x^30); concat([0,0], Vec(serlaplace(asinh(log(x+1)^2)))) \\ _G. C. Greubel_, Oct 28 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Argsinh(Log(x+1)^2) )); [0,0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // _G. C. Greubel_, Oct 28 2018

%K sign

%O 0,3

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

%E Prepended missing a(0)=0, a(1)=0 from _Vaclav Kotesovec_, Nov 02 2013