A012267 - OEIS

%I #27 Sep 08 2022 08:44:38

%S 0,0,2,-6,22,-100,668,-6048,64776,-763488,9918072,-144472680,

%T 2365739880,-42879666960,845124232080,-17930092309920,408038138491200,

%U -9939819541747200,258294825756089760,-7127596576224545760

%N Expansion of e.g.f. arcsin(log(x+1)^2).

%H G. C. Greubel, <a href="/A012267/b012267.txt">Table of n, a(n) for n = 0..418</a> (terms 0..60 from Muniru A Asiru)

%F a(n) ~ (-1)^n * sqrt(2) * n^(n-1) / (exp(1) - 1)^(n - 1/2). - _Vaclav Kotesovec_, Jul 17 2018

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

%p seq(coeff(series(factorial(n)*arcsin(log(x+1)^2), x,n+1),x,n),n=0..20); # _Muniru A Asiru_, Jul 17 2018

%t With[{nn = 30}, CoefficientList[Series[ArcSin[Log[x + 1]^2], {x, 0, nn}], x] Range[0, nn]!] (* _G. C. Greubel_, Oct 28 2018 *)

%o (PARI) x = 'x + O('x^30); concat([0,0], Vec(serlaplace(asin(log(x+1)^2)))) \\ _Michel Marcus_, Jul 17 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Arcsin(Log(x+1)^2) )); [0,0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // _G. C. Greubel_, Oct 25 2018

%K sign

%O 0,3

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

%E a(0) and a(1) inserted by _Sean A. Irvine_, Jul 16 2018

%E Name clarified by _Michel Marcus_, Jul 17 2018