%I #44 Aug 03 2021 08:21:18
%S 1,0,1,-3,12,-60,360,-2520,20160,-181440,1814400,-19958400,239500800,
%T -3113510400,43589145600,-653837184000,10461394944000,
%U -177843714048000,3201186852864000,-60822550204416000,1216451004088320000,-25545471085854720000,562000363888803840000
%N Expansion of e.g.f. cos(i*log(1 + x)), i = sqrt(-1).
%C If the signs are ignored, this is essentially the same as A001710, whose e.g.f. is cos(i*log(1 - x)) = cosh(log(1 - x)).
%C The sequence 0,1,1,3,12,60,... has e.g.f. -Im(sin(i*log(1 - x))) = -sinh(log(1 - x)); the sequence 0,1,-1,3,-12,60,... has e.g.f. Im(sin(i*log(1 + x))) = sinh(log(1 + x)).
%H Vincenzo Librandi, <a href="/A105752/b105752.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f.: cos(i*log(1 + x)), i = sqrt(-1).
%F E.g.f.: 1/2*(1 + x + 1/(1 + x)). - _Sergei N. Gladkovskii_, May 15 2013
%F Let Q(k,x) = 1 + (k+2)*x/(1 - x/(x + 1/Q(k+1,x))), then g.f.: 1 + (Q(0,sqrt(-x)) - 1)*x^2/(2*(sqrt(-x) - x)). - _Sergei N. Gladkovskii_, May 15 2013
%F G.f.: 1 + x^2/2*G(0), where G(k)= 1 + 1/(1 - x*(k+3)/(x*(k+3) - 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 25 2013
%F For n > 1, a(n) = (-1)^n * n! / 2. - _Vaclav Kotesovec_, Feb 25 2014
%F Conjecture: a(n) = Sum_{k=0..n} Stirling1(n, 2*k). - _Benedict W. J. Irwin_, Oct 19 2016
%F E.g.f.: cosh(log(1 + x)). - _Jianing Song_, Apr 06 2019
%t CoefficientList[Series[1/2*(1+x+1/(1+x)), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Feb 25 2014 *)
%o (PARI) x='x+O('x^66); Vec(serlaplace(1/2*(1+x+1/(1+x)))) \\ _Joerg Arndt_, May 15 2013
%Y Cf. A001710.
%K easy,sign
%O 0,4
%A _Paul Barry_, Apr 18 2005