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A166748 E.g.f.: exp(6*arcsin(x)). 4

%I

%S 1,6,36,222,1440,9990,74880,609390,5391360,51798150,539136000,

%T 6060383550,73322496000,951480217350,13198049280000,195053444556750,

%U 3061947432960000,50908949029311750,894088650424320000

%N E.g.f.: exp(6*arcsin(x)).

%C exp(6*arcsin(1/2)) is Aleksandr Gelfond's constant exp(Pi).

%H G. C. Greubel, <a href="/A166748/b166748.txt">Table of n, a(n) for n = 0..445</a>

%H A. R. Povolotsky et al., <a href="http://groups.google.com/group/sci.math.symbolic/browse_thread/thread/1b5e3b73f1d93b8d/88d5669fc28f3bf8?hl=en#88d5669fc28f3bf8">With regards to OEIS A166748</a>, sci.math.symbolic usenet group, 2009

%F Contribution from _Alexander R. Povolotsky_, Oct 24 2009: (Start)

%F a(n+2) = (n^2+36)*a(n), a(0)=1, a(1)=6.

%F The above recurrence leads to

%F a(n) = (3*2^n*gamma(-3*i+n/2)*gamma(3*i+n/2)*(cos((n*Pi)/2)+i*sin((n*Pi)/2))*sinh(((6-i*n)*Pi)/2))/Pi where "i" is imaginary unit. (End)

%F a(n) = 3*2^(n-1)*(exp(3*Pi)-(-1)^n*exp(-3*Pi))*|Gamma(n/2+3i)|^2/Pi. - _R. J. Mathar_ and _M. F. Hasler_, Oct 25 2009

%F a(n) ~ 6 * (exp(3*Pi) - (-1)^n*exp(-3*Pi)) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Nov 06 2014

%t Round[Table[3*2^(n-1)*(E^(3*Pi)-(-1)^n*E^(-3*Pi))*Abs[Gamma[n/2+3*I]]^2/Pi,{n,0,20}]] (* _Vaclav Kotesovec_, Nov 06 2014 *)

%t CoefficientList[Series[Exp[6*ArcSin[x]], {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Nov 06 2014 *)

%o (PARI) A166748(n)=round(norm(gamma(n/2+3*I))/Pi*if(n%2,cosh(3*Pi),sinh(3*Pi))*3<<n) \\ [_M. F. Hasler_, Oct 25 2009]

%o (PARI) a(n)=polcoeff(exp(6*asin(x)),n)*n!

%o (PARI) a(n)=(1+5*(n%2))*prod(k=0,n\2-1,(2*k+n%2)^2+36) [_Jaume Oliver Lafont_, Oct 28 2009]

%Y Cf. A166741, A006228, A039661.

%K nonn

%O 0,2

%A _Jaume Oliver Lafont_, Oct 21 2009

%E Minor edits by _Vaclav Kotesovec_, Nov 06 2014

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