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%I #16 Jan 05 2017 17:15:06
%S 1,16,3072,737280,1321205760,951268147200,2009078326886400,
%T 265928913086054400,44931349155019751424000,
%U 109991942731488351485952000,668751011807449177034588160000,2471703739640332158319837839360000
%N A sequence related to the period T of a simple gravity pendulum for arbitrary amplitudes.
%C The period T of a simple gravity pendulum for arbitrary amplitudes is given by a complicated formula, see A223067. The Taylor series expansion of T as a function of the angular displacement phi leads for the denominators of the even powers of phi to the sequence given above and for the numerators to A223067.
%e T = 2*Pi*sqrt(L/g) * (1 + (1/16)*phi^2 + (11/3072)*phi^4 + (173/737280)*phi^6 + ... ).
%p nmax:=11: f := series(1/((Pi/4)*(1+cos(phi/2))/EllipticK((1-cos(phi/2))/(1+cos(phi/2)))), phi, 2*nmax+1): for n from 0 to nmax do a(n):= denom(coeff(f, phi, 2*n)) od: seq(a(n), n=0..nmax); # _Johannes W. Meijer_, Jan 05 2017
%t s = Series[EllipticK[Sin[t/2]^2 ], {t, 0, 50}]; CoefficientList[2*s, t^2] // Denominator (* _Jean-François Alcover_, Oct 07 2014 *)
%Y Cf. A223067 (numerators), A019692 (2*Pi).
%Y Cf. A280442, A280443.
%K nonn,easy,frac
%O 0,2
%A _Johannes W. Meijer_, Mar 14 2013