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 A223068 A sequence related to the period T of a simple gravity pendulum for arbitrary amplitudes. 3
 1, 16, 3072, 737280, 1321205760, 951268147200, 2009078326886400, 265928913086054400, 44931349155019751424000, 109991942731488351485952000, 668751011807449177034588160000, 2471703739640332158319837839360000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. LINKS EXAMPLE T = 2*Pi*sqrt(L/g) * (1 + (1/16)*phi^2 + (11/3072)*phi^4 + (173/737280)*phi^6 + ... ). MAPLE 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 MATHEMATICA s = Series[EllipticK[Sin[t/2]^2 ], {t, 0, 50}]; CoefficientList[2*s, t^2] // Denominator (* Jean-François Alcover, Oct 07 2014 *) CROSSREFS Cf. A223067 (numerators), A019692 (2*Pi). Cf. A280442, A280443. Sequence in context: A123282 A091160 A049030 * A051551 A307930 A289703 Adjacent sequences:  A223065 A223066 A223067 * A223069 A223070 A223071 KEYWORD nonn,easy,frac AUTHOR Johannes W. Meijer, Mar 14 2013 STATUS approved

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Last modified December 1 11:29 EST 2021. Contains 349429 sequences. (Running on oeis4.)