

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
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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

Table of n, a(n) for n=0..11.


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((1cos(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 (* JeanFrançois 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



