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%I #14 Nov 06 2014 15:02:21
%S 1,2,4,10,32,130,640,3770,25600,199810,1740800,16983850,181043200,
%T 2122981250,26794393600,367275756250,5358878720000,84106148181250,
%U 1393308467200000,24643101417106250,457005177241600000
%N E.g.f.: exp(2*arcsin(x)).
%C exp(2*arcsin(1)) is Aleksandr Gelfond's constant.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Gelfond's_constant">Gelfond's constant</a>
%F a(n) ~ 2 * n^(n-1) * (exp(Pi) - (-1)^n/exp(Pi)) / exp(n). - _Vaclav Kotesovec_, Aug 04 2014
%F From _Vaclav Kotesovec_, Nov 06 2014: (Start)
%F a(n) = (n^2 - 4*n + 8)*a(n-2).
%F a(n) = 2^(n-1) * (exp(Pi)-(-1)^n*exp(-Pi)) * GAMMA(n/2-I) * GAMMA(n/2+I) / Pi.
%F (End)
%p seq(simplify(2^(n-1) * (cosh(Pi)*(1-(-1)^n) + sinh(Pi)*(1+(-1)^n)) * GAMMA((1/2)*n-I)*GAMMA((1/2)*n+I) / Pi), n=0..20); # _Vaclav Kotesovec_, Nov 06 2014
%t CoefficientList[Series[E^(2*ArcSin[x]), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Aug 04 2014 *)
%t FullSimplify[Table[2^(n-1) * (E^(Pi)-(-1)^n*E^(-Pi)) * Gamma[n/2-I] * Gamma[n/2+I] / Pi,{n,0,20}]] (* _Vaclav Kotesovec_, Nov 06 2014 *)
%o (PARI) for (n=0,25,print(polcoeff(exp(2*asin(x)),n)*n!,","))
%Y Cf. A006228, A039661, A166748.
%K nonn
%O 0,2
%A _Jaume Oliver Lafont_, Oct 21 2009
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