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User: Dan T. Abell

Dan T. Abell has authored 2 sequences.

A189805 Numerators of coefficients in the series expansion of ((2 - m) EllipticK(m) - 2 EllipticE(m))/(Pi * m).

0, 1, 3, 75, 245, 6615, 22869, 1288287, 4601025, 265939245, 969738055, 28510298817, 105468168351, 3138933581875, 11734782467625, 1409850293610375, 5313200518272825, 642897262711011825, 2438232652696561875, 74176455589813182375, 282762879579212657625

Offset: 0

Dan T. Abell, Apr 27 2011

This combination of elliptic functions appears in the expression for the vector potential generated by a circular loop of current.

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

G. C. Greubel, Table of n, a(n) for n = 0..830

Cf. A189806.

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

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

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

Original entry on oeis.org

1, 16, 64, 2048, 8192, 262144, 1048576, 67108864, 268435456, 17179869184, 68719476736, 2199023255552, 8796093022208, 281474976710656, 1125899906842624, 144115188075855872, 576460752303423488, 73786976294838206464, 295147905179352825856, 9444732965739290427392, 37778931862957161709568

Offset: 0

Dan T. Abell, Apr 28 2011

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.

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

G. C. Greubel, Table of n, a(n) for n = 0..830

Cf. A189805, A090739, A007814.

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

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

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User: Dan T. Abell

A189805 Numerators of coefficients in the series expansion of ((2 - m) EllipticK(m) - 2 EllipticE(m))/(Pi * m).

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