cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A261975 Expansion of elliptic_E / elliptic_K in powers of q.

Original entry on oeis.org

1, -8, 48, -224, 864, -2928, 9024, -25792, 69312, -176936, 432288, -1016736, 2312832, -5107504, 10983552, -23060544, 47373696, -95401872, 188637936, -366744160, 701930304, -1324016896, 2463662016, -4526174784, 8216376576, -14747939768, 26191413024, -46048199360
Offset: 0

Views

Author

Joerg Arndt, Sep 07 2015

Keywords

Links

Crossrefs

Cf. A004018 (elliptic K(q)), A194094 (elliptic E(q)), A115977 (elliptic k(q)^2).
Cf. A261975 (E/K), A261977 ((E/K)^(1/2)), A261978 ((E/K)^(1/4)).
Cf. A261976 (K/E), A261979 ((K/E)^(1/2)), A261980 ((K/E)^(1/4)).

Programs

  • Mathematica
    nmax = 30; dtheta = D[Normal[Series[EllipticTheta[3, 0, x], {x, 0, nmax}]], x]; CoefficientList[Series[(EllipticTheta[4, 0, x]^4*EllipticTheta[3, 0, x] + 4*x*dtheta)/EllipticTheta[3, 0, x]^5, {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2018 *)

Formula

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

A261976 Expansion of elliptic_K / elliptic_E in powers of q.

Original entry on oeis.org

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

Views

Author

Joerg Arndt, Sep 07 2015

Keywords

Crossrefs

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)).

Formula

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

Views

Author

Joerg Arndt, Sep 07 2015

Keywords

Crossrefs

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).

Formula

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

Views

Author

Joerg Arndt, Sep 07 2015

Keywords

Crossrefs

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).

Formula

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

Views

Author

Joerg Arndt, Sep 07 2015

Keywords

Crossrefs

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).

Formula

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

Views

Author

Joerg Arndt, Sep 07 2015

Keywords

Crossrefs

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).

Formula

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

A193219 Expansion of sqrt((2/Pi)*elliptic_E(k)) in powers of q.

Original entry on oeis.org

1, -2, 8, -16, 18, -32, 112, -192, 0, 62, 1840, -3312, -8320, 16480, 71840, -137280, -522174, 1011392, 4107960, -7945008, -32457600, 62909120, 261338416, -506930112, -2129035776, 4133297534, 17531850576, -34058050240, -145663683072, 283125653280, 1219649036576, -2371704375168, -10281070960128, 20000146662464, 87178011852896
Offset: 0

Views

Author

Joerg Arndt, Aug 26 2011

Keywords

Comments

Let s = 16*q*(E1*E4^2/E2^3)^8 where Ek = Product_{n>=1} (1-q^(k*n)) (s=k^2 where k is elliptic k), then the g.f. is sqrt(hypergeom([-1/2, +1/2], [+1], s)) (expansion of sqrt((2/Pi)*elliptic_E(k)) in powers of q).
The corresponding sequence for sqrt((2/Pi)*elliptic_K(k)) is A000122.

Examples

			sqrt(E(k(q))) = 1 - 2*q + 8*q^2 - 16*q^3 + 18*q^4 - 32*q^5 + 112*q^6 - 192*q^7 +- ...

Links

Crossrefs

Cf. A194094 (elliptic_E(k(q))), A004018 (elliptic_K(k(q))), A000122 (sqrt(elliptic_K(k(q)))=Theta3(q)), A115977 (elliptic k(q)^2).
Cf. A072558, A073007.

Programs

  • Mathematica
    CoefficientList[Series[Sqrt[(2/Pi) EllipticE[InverseEllipticNomeQ[q]]], {q, 0, 50}], q] (* Jan Mangaldan, Dec 07 2021 *)
nmax = 30; dtheta = D[Normal[Series[EllipticTheta[3, 0, x], {x, 0, nmax}]], x]; CoefficientList[Series[Sqrt[(EllipticTheta[4, 0, x]^4*EllipticTheta[3, 0, x] + 4*x*dtheta)/EllipticTheta[3, 0, x]^3], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 15 2023 *)

Formula

From Vaclav Kotesovec, Nov 16 2023: (Start)
abs(a(n)) ~ c * d^n / n^(3/2), where
d = 1/sqrt(A072558) = sqrt(A073007) = 3.0477902637682959365706804198489438625220426001497960504423261561153885844...
c = 0.60315114232684465914106139794838284733424313832900503234838172483814652... if n is even and
c = 0.38688142678580145044658710898009855553630625532976316366806686926256857... if n is odd. (End)
Showing 1-7 of 7 results.

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