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.

A051273 Expansion of q^(-1/3) * b(q) * c(q) / a(q)^2 in powers of q where a(), b(), c() are cubic AGM theta functions.

Original entry on oeis.org

3, -42, 393, -3240, 24999, -184740, 1325679, -9312408, 64364025, -439225086, 2966629452, -19868187384, 132119675241, -873278632080, 5742216378024, -37587341460600, 245063740036086, -1592173816624290, 10311978807488160
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Coefficients in a certain q-series associated with a failed attempt to explain a mysterious entry in a Ramanujan notebook.

Examples

			G.f. = 3 - 42*x + 393*x^2 - 3240*x^3 + 24999*x^4 - 184740*x^5 + ...
G.f. = 3*q - 42*q^4 + 393*q^7 - 3240*q^10 + 24999*x^13 - 184740*q^16 + ...

References

  • B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 179, Eq. 13.23.

Links

Crossrefs

Cf. A004016, A058092, A258941, A258942, A328785.

Programs

  • Mathematica
    a[0] = 3; a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[3*(QPochhammer[ x + A]*(QPochhammer[x^3 + A]^2/(QPochhammer[x + A]^3 + 9*x * QPochhammer[ x^9 + A]^3)))^2, n]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 06 2015, adapted from PARI *)
CoefficientList[Series[3 * (QPochhammer[x,x] * QPochhammer[x^3,x^3]^2 / (QPochhammer[x,x]^3 + 9*x*QPochhammer[x^9,x^9]^3))^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 07 2015 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 3 * (eta(x + A) * eta(x^3 + A)^2 / (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3))^2, n))}; /* Michael Somos, Aug 07 2006 */

Formula

Expansion of 3*(eta(q)*eta(q^3))^2/(theta[A_2](q)^2*q^(1/3)) in powers of q.
a(n) ~ (-1)^n * c * n * exp(Pi*n/sqrt(3)), where c = 3 * A258942^2 = 192 * exp(Pi/(3*sqrt(3))) * Pi^5 / Gamma(1/6)^6 = 3.6159115405362166049256277... . - Vaclav Kotesovec, Nov 07 2015, updated Nov 14 2015
a(n) = 3*A328785(n). - Michael Somos, Nov 02 2019

Extensions

Corrected and extended by Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 15 2000

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

A115784 Expansion of b(q) / a(q) in powers of q of cubic AGM theta function.

Original entry on oeis.org

1, -9, 54, -324, 1989, -12204, 74844, -459072, 2815830, -17271468, 105938118, -649793448, 3985642908, -24446767374, 149949318096, -919745243064, 5641448209173, -34602992662356, 212244632371188, -1301846473509156, 7985145356345268, -48978545212087776
Offset: 0

Views

Author

Michael Somos, Jan 31 2006

Keywords

Examples

			1 - 9*q + 54*q^2 - 324*q^3 + 1989*q^4 - 12204*q^5 + 74844*q^6 - 459072*q^7 + ...

Links

Crossrefs

Cf. A004016, A005928, A215690, A258942.

Programs

  • Mathematica
    QP = QPochhammer;  s = QP[q]^3/(QP[q]^3 + 9*q*QP[q^9]^3) + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
eta[q_] := q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[eta[q]^3/ (eta[q]^3 + 9*eta[q^9]^3), {q,0,n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Feb 11 2018 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 / (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3), n))}

Formula

Expansion of eta(q)^3 / (eta(q)^3 + 9 * eta(q^9)^3) in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 - u*v)^3 - (1 - u^3) * (1 - v^3).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (1 + 2*u)^3 * v^3 - 9 * u * (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) = (1 + 2*u1) * (1 + 2*u2) * u3*u6 - 3 * (u1 + u2 + u1*u2).
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 / (1 + 9 * x * (Product_{k>0} (1 - x^(9*k)) / (1 - x^k))^3).
Convolution inverse is A215690. Convolution with A004016 is A005928.
a(n) ~ (-1)^n * 8 * sqrt(3) * Pi^(5/2) * exp(Pi*n/sqrt(3)) / Gamma(1/6)^3. - Vaclav Kotesovec, Nov 14 2015
Showing 1-3 of 3 results.

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