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 1-3 of 3 results.

A089602 Expansion of L(x)^(1/4), where L(x) = o.g.f. for A053175.

Original entry on oeis.org

1, 2, 14, 132, 1446, 17340, 220524, 2919240, 39761094, 553080044, 7818246436, 111929301688, 1618972088028, 23616939932376, 346986771074328, 5129262870441360, 76223971339368006, 1137977844577647948, 17058656523389665268, 256642078290095158360, 3873624648355421605492
Offset: 0

Views

Author

Vladeta Jovovic, Dec 30 2003

Keywords

Links

Crossrefs

Cf. A053175, A090004.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[(EllipticK[(8*x/(1 - 8*x))^2]/((1 - 8*x)*Pi/2))^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 26 2019 *)
  • PARI
    Vec(1/agm(1,1-16*x+O(x^66))^(1/4)) \\ Joerg Arndt, Aug 14 2013

Formula

G.f.: (2*EllipticK(8*x/(1-8*x))/((1-8*x)*Pi))^(1/4).
a(n) ~ 2^(4*n - 7/4) / (Pi^(1/4) * n * log(n)^(3/4)) * (1 - (gamma/2 + log(2)) / log(n) + (3*gamma^2/8 + 3*log(2)*gamma/2 + 3*log(2)^2/2 - Pi^2/16) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

A323385 Expansion of AGM(1,1-16*x) (where AGM denotes the arithmetic-geometric mean).

Original entry on oeis.org

1, -8, -16, -128, -1344, -15872, -199680, -2613248, -35148800, -482500608, -6730072064, -95094702080, -1358152794112, -19573573681152, -284284750397440, -4156674357067776, -61133523873169408, -903754859816157184, -13421680957337894912, -200140704802846801920
Offset: 0

Views

Author

Seiichi Manyama, Jan 13 2019

Keywords

Links

Crossrefs

Convolution inverse of A053175.
Cf. A060691, A089602, A090004.

Programs

  • Maple
    R:= Pi*(1-8*x)/(2*EllipticK(-8*x/(1-8*x))):
S:= series(R,x,31):
seq(coeff(S,x,j),j=0..30); # Robert Israel, Jan 13 2019
  • Mathematica
    CoefficientList[Series[Pi*(1 - 16*x) / (2*EllipticK[1 - 1/(1 - 16*x)^2]), {x, 0, 25}], x] (* Vaclav Kotesovec, Sep 28 2019 *)
  • PARI
    N=66; x='x+O('x^N); Vec(agm(1, 1-16*x))

Formula

a(n) = 2^n * A060691(n).
a(n) ~ -Pi * 2^(4*n-1) / (n * log(n)^2) * (1 - (2*gamma + 4*log(2))/log(n) + (3*gamma^2 + 12*log(2)*gamma + 12*log(2)^2 - Pi^2/2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

A228156 Expansion of sqrt((1+4*x)/AGM(1+4*x,1-4*x)) where AGM denotes the arithmetic-geometric mean.

Original entry on oeis.org

1, 2, 0, 8, 2, 68, 32, 720, 464, 8480, 6656, 106368, 95912, 1390928, 1392512, 18734144, 20371650, 257955716, 300101760, 3613109008, 4448177412, 51302395528, 66289160512, 736588435360, 992578330048, 10674012880512, 14924667774976, 155890890782720, 225244659392784, 2291995151532576, 3410654921389824
Offset: 0

Views

Author

Joerg Arndt, Aug 14 2013

Keywords

Comments

Convolution square is A092266.

Links

Crossrefs

Cf. A092266 (1+4*x)/AGM(1+4*x,1-4*x).
Cf. A081085 1/AGM(1,1-8*x), A053175 1/AGM(1,1-16*x), A090004 1/AGM(1,1-16*x)^(1/2), A089602 1/AGM(1,1-16*x)^(1/4).

Programs

  • Mathematica
    CoefficientList[Series[Sqrt[2*(1 + 4*x)*EllipticK[1 - (1 + 4*x)^2/(1 - 4*x)^2] / (Pi*(1 - 4*x))], {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 27 2019 *)
  • PARI
    Vec( 1/agm(1,(1-4*x)/(1+4*x)+O(x^66))^(1/2) ) \\ Joerg Arndt, Aug 14 2013

Formula

a(n) ~ 2^(2*n - 1/2) / (n*sqrt(Pi*log(n))) * (1 - (gamma + 3*log(2)) / (2*log(n)) + (3*gamma^2/8 + 9*gamma*log(2)/4 + 27*log(2)^2/8 - 1/16*Pi^2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019
Showing 1-3 of 3 results.

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