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.

A258941 Convolution inverse of A058537.

Original entry on oeis.org

1, -7, 41, -253, 1555, -9532, 58463, -358600, 2199546, -13491360, 82752059, -507576937, 3113328401, -19096245457, 117130782240, -718445946527, 4406737223117, -27029636742811, 165791883077354, -1016918901125280, 6237482995373629, -38258895644996020
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 07 2015

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Convolution square is A328785. - Michael Somos, Nov 02 2019

Examples

			G.f. = 1 - 7*x + 41*x^2 - 253*x^3 + 1555*x^4 - 9532*x^5 + ... - _Michael Somos_, Nov 02 2019
G.f. = q - 7*q^7 + 41*q^13 - 253*q^19 + 1555*q^25 - 9532*q^31 + ... - _Michael Somos_, Nov 02 2019

Links

Crossrefs

Cf. A058537, A051273, A058092, A115784, A258942, A328785.

Programs

  • Mathematica
    CoefficientList[Series[QPochhammer[x, x] * QPochhammer[x^3, x^3]^2 / (QPochhammer[x, x]^3 + 9*x*QPochhammer[x^9, x^9]^3), {x, 0, 50}], x]
eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/6)* eta[q]*eta[q^9]^2/(eta[q]^3 + 9*eta[q^9]^3), {q,0,60}], q] (* G. C. Greubel, Jun 22 2018 *)
a[ n_] := SeriesCoefficient[ QPochhammer[x] QPochhammer[x^3]^2 / (QPochhammer[x]^3 + 9 x QPochhammer[x^9]^3), {x, 0, n}]; (* Michael Somos, Nov 02 2019 *)
  • PARI
    q='q+O('q^50); A = eta(q)*eta(q^3)^2/(eta(q)^3 + 9*q*eta(q^9)^3); Vec(A) \\ G. C. Greubel, Jun 22 2018

Formula

a(n) ~ (-1)^n * c * exp(Pi*n/sqrt(3)), where c = A258942 = 8*exp(Pi/(6*sqrt(3))) * Pi^(5/2) / Gamma(1/6)^3 = 1.09786330972731096865822482325074133091288... . - Vaclav Kotesovec, Nov 14 2015
Expansion of q^(-1/6)* eta[q]*eta[q^9]^2/(eta[q]^3 + 9*eta[q^9]^3) in powers of q. - G. C. Greubel, Jun 22 2018
Expansion of q^(-1/6) * 3^(-1/2) * sqrt(b(q)*c(q))/a(q) in powers of q where a(), b(), c() are cubic AGM functions. - Michael Somos, Nov 02 2019

A215690 Expansion of a(q) / b(q) in powers of q where a(), b() are cubic AGM theta functions.

Original entry on oeis.org

1, 9, 27, 81, 198, 459, 972, 1989, 3861, 7290, 13284, 23679, 41148, 70218, 117504, 193671, 314262, 503415, 796068, 1244988, 1925910, 2950668, 4478328, 6739497, 10059228, 14901471, 21914442, 32011119, 46456272, 67010679, 96093864, 137039922, 194395221
Offset: 0

Views

Author

Michael Somos, Aug 20 2012

Keywords

Examples

			1 + 9*q + 27*q^2 + 81*q^3 + 198*q^4 + 459*q^5 + 972*q^6 + 1989*q^7 + 3861*q^8 + ...

References

  • O. Kolberg, The coefficients of j(tau) modulo powers of 3, Acta Univ. Bergen., Series Math., Arbok for Universitetet I Bergen, Mat.-Naturv. Serie, 1962 No. 16, pp. 1-7. See u, page 1.

Links

Crossrefs

Cf. A004016, A005928, A058091, A115784, A121589.

Programs

  • Mathematica
    QP = QPochhammer; s = 1 + 9*q*(QP[q^9]/QP[q])^3 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 + 9 * x * (eta(x^9 + A) / eta(x + A))^3, n))}

Formula

Expansion of 1 + 3 * c(q^3) / b(q) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of 1 + 9 * (eta(q^9) / eta(q))^3 in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u * v - 1)^3 - (u^3 - 1) * (v^3 - 1).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u + 2)^3 - 9 * v^3 * (1 + u + u^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u1 + 2) * (u2 + 2) - 3 * (1 + u1 + u2) * u3*u6.
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A058091.
G.f.: 1 + 9 * x * (Product_{k>0} (1 - x^(9*k)) / (1 - x^k))^3.
a(n) = 9 * A121589(n) unless n=0.
Convolution inverse is A115784. Convolution with A005928 is A004016.
a(n) ~ exp(4*Pi*sqrt(n)/3) / (3 * sqrt(6) * n^(3/4)). - Vaclav Kotesovec, Nov 14 2015

A258942 Decimal expansion of a constant related to A258941.

Original entry on oeis.org

1, 0, 9, 7, 8, 6, 3, 3, 0, 9, 7, 2, 7, 3, 1, 0, 9, 6, 8, 6, 5, 8, 2, 2, 4, 8, 2, 3, 2, 5, 0, 7, 4, 1, 3, 3, 0, 9, 1, 2, 8, 8, 0, 8, 7, 3, 8, 9, 3, 6, 3, 0, 4, 5, 7, 9, 1, 6, 4, 9, 8, 2, 5, 9, 9, 4, 0, 9, 2, 7, 3, 8, 5, 2, 4, 9, 4, 3, 2, 4, 8, 1, 7, 1, 8, 3, 6, 1, 6, 0, 2, 3, 7, 2, 1, 4, 3, 1, 0, 1, 7, 7, 4, 8, 1
Offset: 1

Views

Author

Vaclav Kotesovec, Nov 07 2015

Keywords

Examples

			1.09786330972731096865822482325074133091288087389363045791649825994...

Crossrefs

Cf. A051273, A115784, A258941.

Programs

  • Maple
    evalf(8*exp(Pi/(6*sqrt(3))) * Pi^(5/2) / GAMMA(1/6)^3, 120); # Vaclav Kotesovec, Nov 14 2015
  • Mathematica
    RealDigits[8*E^(Pi/(6*Sqrt[3]))*Pi^(5/2)/Gamma[1/6]^3, 10, 105][[1]] (* Vaclav Kotesovec, Nov 14 2015 *)

Formula

Equals limit n->infinity A258941(n) * (-1)^n / exp(Pi*n/sqrt(3)).
Equals 8*exp(Pi/(6*sqrt(3))) * Pi^(5/2) / Gamma(1/6)^3. - Vaclav Kotesovec, Nov 14 2015
Showing 1-3 of 3 results.

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