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%I #15 Sep 08 2022 08:44:38
%S 0,0,2,-3,12,-40,478,-3759,40152,-387864,5203962,-70961715,1142838180,
%T -18621542160,342700299030,-6622936396335,140637395893680,
%U -3129194141386320,74723900156463090,-1873298738386231875
%N Expansion of e.g.f. arctanh(arcsin(x) * log(x+1)).
%H G. C. Greubel, <a href="/A012312/b012312.txt">Table of n, a(n) for n = 0..426</a>
%e E.g.f. = 2*x^2/2! - 3*x^3/3! + 12*x^4/4! - 40*x^5/5! + ...
%p seq(coeff(series(factorial(n)*arctanh(arcsin(x)*log(x+1)),x,n+1), x, n), n = 0 .. 20); # _Muniru A Asiru_, Oct 25 2018
%t With[{nn = 30}, CoefficientList[Series[ArcTanh[ArcSin[x] Log[x + 1]], {x, 0, nn}], x] Range[0, nn]!] (* _G. C. Greubel_, Oct 25 2018 *)
%o (PARI) x='x+O('x^30); concat([0,0], Vec(serlaplace(atanh(asin(x)* log(x+1)) ))) \\ _G. C. Greubel_, Oct 25 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Argtanh(Arcsin(x)*Log(x+1)) )); [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 and title improved by _Sean A. Irvine_, Jul 17 2018
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