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 A189806 Denominators of coefficients in the series expansion of ((2 - m) EllipticK(m) - 2 EllipticE(m))/(Pi * m). 2

%I #14 Jan 17 2018 03:33:24

%S 1,16,64,2048,8192,262144,1048576,67108864,268435456,17179869184,

%T 68719476736,2199023255552,8796093022208,281474976710656,

%U 1125899906842624,144115188075855872,576460752303423488,73786976294838206464,295147905179352825856,9444732965739290427392,37778931862957161709568

%N Denominators of coefficients in the series expansion of ((2 - m) EllipticK(m) - 2 EllipticE(m))/(Pi * m).

%C This combination of elliptic functions appears in the expression for the vector potential generated by a circular loop of current. The denominators are powers of 2. The base-2 logarithm of the denominators increments in pattern related to A090739. That latter sequence begins 3,4,3,5,3,4,3,6. Add 2 to each entry; thus, 5,6,5,7,5,6,5,8. Duplicate each entry; thus, 5,5,6,6,5,5,7,7,5,5,6,6,5,5,8,8. Now insert a 2 at the beginning and between each entry; thus, 2,5,2,5,2,6,2,6,2,5,2,5,2, 7,2,7,2,5,2,5,2,6,2,6,2,5,2,5,2,8,2,8. Finally, prepend a 4; thus 4,2,5,2,5,2,6,2,6,2,5,2,5,2,7,2,7,2,5,2,5,2,6,2,6,2,5,2,5,2,8,2,8. This yields the pattern of increments in the base-2 logarithm of the denominators. See also the construction of the ruler sequence A007814.

%D J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, third edition, 1999, eq.(5.37).

%H G. C. Greubel, <a href="/A189806/b189806.txt">Table of n, a(n) for n = 0..830</a>

%F a(n) is the denominator of the fraction ((2n-1)!!)^2/(2^(2n+1)*(n-1)!*(n+1)!).

%t Denominator[CoefficientList[Series[((2-m)EllipticK[m]-2EllipticE[m])/m,{m,0,20}]/Pi,m]]

%Y Cf. A189805, A090739, A007814.

%K nonn,frac

%O 0,2

%A _Dan T. Abell_, Apr 28 2011

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Last modified August 14 13:35 EDT 2024. Contains 375165 sequences. (Running on oeis4.)