A261975 -id:A261975 - cp's OEIS frontend

A261976 Expansion of elliptic_K / elliptic_E in powers of q.

1, 8, 16, -32, -96, 368, 960, -3392, -8896, 31528, 82656, -292704, -767616, 2719024, 7130496, -25257408, -66235776, 234616720, 615265872, -2179359392, -5715218752, 20244124928, 53088812352, -188048196544, -493143336192, 1746784492472, 4580821023328, -16225925666624

Offset: 0

Joerg Arndt, Sep 07 2015

sign

Cf. A004018 (elliptic K(q)), A194094 (elliptic E(q)), A115977 (elliptic k(q)^2).
Cf. A261979 ((K/E)^(1/2)), A261980 ((K/E)^(1/4)).
Cf. A261975 (E/K), A261977 ((E/K)^(1/2)), A261978 ((E/K)^(1/4)).

G.f.: T3^5 / (T4^4 * T3 + 4*q * d/dq T3) where T3 = theta_3(q) and T4 = theta_4(q).

A261977 Expansion of (elliptic_E / elliptic_K)^(1/2) in powers of q.

Original entry on oeis.org

1, -4, 16, -48, 112, -248, 576, -1248, 2272, -3988, 8672, -18192, 23616, -23000, 100992, -304032, 41152, 970552, 1972816, -11299824, -9904096, 80729472, 95978688, -676487328, -755649408, 5483063076, 6371808608, -45452602080, -53224627584, 378628636264, 449486486400, -3179963494272

Offset: 0

Joerg Arndt, Sep 07 2015

sign

Cf. A261975 (E/K), A261978 ((E/K)^(1/4)).
Cf. A261976 (K/E), A261979 ((K/E)^(1/2)), A261980 ((K/E)^(1/4)).
Cf. A004018 (elliptic K(q)), A194094 (elliptic E(q)), A115977 (elliptic k(q)^2).

See A261975 for the g.f. for E/K.

A261978 Expansion of (elliptic_E / elliptic_K)^(1/4) in powers of q.

Original entry on oeis.org

1, -2, 6, -12, 14, -24, 84, -144, -42, 130, 1656, -3036, -9036, 17784, 76944, -147984, -591274, 1147068, 4784922, -9277164, -38983272, 75690528, 322116804, -625832880, -2687394012, 5224589254, 22613921832, -43985741688, -191670898032, 372970548504, 1634759644944, -3182191744320

Offset: 0

Joerg Arndt, Sep 07 2015

sign

Cf. A261975 (E/K), A261977 ((E/K)^(1/2)).
Cf. A261976 (K/E), A261979 ((K/E)^(1/2)), A261980 ((K/E)^(1/4)).
Cf. A004018 (elliptic K(q)), A194094 (elliptic E(q)), A115977 (elliptic k(q)^2).

See A261975 for the g.f. for E/K.

A261979 Expansion of (elliptic_K / elliptic_E)^(1/2) in powers of q.

Original entry on oeis.org

1, 4, 0, -16, 16, 120, -128, -928, 1056, 7572, -8960, -63408, 77248, 540504, -672000, -4665824, 5888832, 40656072, -51913728, -356835664, 459890400, 3150052992, -4090609024, -27939033312, 36509767552, 248772971228, -326815190784, -2222432164768, 2932886151552, 19910399315736

Offset: 0

Joerg Arndt, Sep 07 2015

sign

Cf. A261975 (E/K), A261977 ((E/K)^(1/2)), A261978 ((E/K)^(1/4)).
Cf. A261976 (K/E), A261980 ((K/E)^(1/4)).
Cf. A004018 (elliptic K(q)), A194094 (elliptic E(q)), A115977 (elliptic k(q)^2).

See A261976 for the g.f. for K/E.

A261980 Expansion of (elliptic_K / elliptic_E)^(1/4) in powers of q.

Original entry on oeis.org

1, 2, -2, -4, 14, 24, -92, -176, 694, 1342, -5480, -10612, 44532, 86408, -369328, -717616, 3109078, 6046724, -26473950, -51523620, 227477656, 442950880, -1969014572, -3835720208, 17147433572, 33415180858, -150096433272, -292574352808, 1319581377424, 2572787175656, -11644937717296

Offset: 0

Joerg Arndt, Sep 07 2015

sign

Cf. A261975 (E/K), A261977 ((E/K)^(1/2)), A261978 ((E/K)^(1/4)).
Cf. A261976 (K/E), A261979 ((K/E)^(1/2)).
Cf. A004018 (elliptic K(q)), A194094 (elliptic E(q)), A115977 (elliptic k(q)^2).

See A261976 for the g.f. for K/E.

A328127 G.f.: E(4sqrt(x)) / K(4sqrt(x)), where E(), K() are complete elliptic integrals.

Original entry on oeis.org

1, -8, -16, -128, -1312, -15104, -186112, -2398208, -31898176, -434421248, -6025687552, -84808699904, -1207939190272, -17375932633088, -252046328713216, -3682284573851648, -54130292542567552, -800036763837307904, -11880834659028677632, -177181827571092267008

Offset: 0

Vaclav Kotesovec, Oct 04 2019

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Vaclav Kotesovec, Table of n, a(n) for n = 0..825
Vaclav Kotesovec, Graph - the asymptotic ratio (30000 terms)
Eric Weisstein's MathWorld, Complete Elliptic Integral of the First Kind.
Eric Weisstein's MathWorld, Complete Elliptic Integral of the Second Kind.

Cf. A010370, A054474, A261975, A328128.

seq(coeff(series(EllipticE(4*sqrt(x))/EllipticK(4*sqrt(x)), x, 21), x, n), n = 0..20);

CoefficientList[Series[EllipticE[16*x]/EllipticK[16*x], {x, 0, 20}], x]

a(n) ~ -2^(4*n+1) / (n * log(n)^2) * (1 - (2*gamma + 8*log(2)) / log(n) + (3*gamma^2 + 24*log(2)*gamma + 48*log(2)^2 - Pi^2/2) / log(n)^2 + (-4*gamma^3 + 2*gamma*Pi^2 - 48*gamma^2*log(2) + 8*Pi^2*log(2) - 192*gamma*log(2)^2 - 256*log(2)^3 - 8*Zeta(3)) / log(n)^3 + (5*gamma^4 - 5*gamma^2*Pi^2 + Pi^4/12 + 80*gamma^3*log(2) - 40*gamma*Pi^2*log(2) + 480*gamma^2*log(2)^2 - 80*Pi^2*log(2)^2 + 1280*gamma*log(2)^3 + 1280*log(2)^4 + 40*gamma*Zeta(3) + 160*log(2)*Zeta(3)) / log(n)^4), where gamma is the Euler-Mascheroni constant A001620.

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A261976 Expansion of elliptic_K / elliptic_E in powers of q.

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A261977 Expansion of (elliptic_E / elliptic_K)^(1/2) in powers of q.

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A261978 Expansion of (elliptic_E / elliptic_K)^(1/4) in powers of q.

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A261979 Expansion of (elliptic_K / elliptic_E)^(1/2) in powers of q.

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A261980 Expansion of (elliptic_K / elliptic_E)^(1/4) in powers of q.

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A328127 G.f.: E(4*sqrt(x)) / K(4*sqrt(x)), where E(), K() are complete elliptic integrals.

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Formula

A328127 G.f.: E(4sqrt(x)) / K(4sqrt(x)), where E(), K() are complete elliptic integrals.