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 1-4 of 4 results.

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

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

Views

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

A058537 McKay-Thompson series of class 18b for the Monster group.

Original entry on oeis.org

1, 7, 8, 22, 42, 63, 106, 190, 267, 428, 652, 932, 1367, 2017, 2774, 3950, 5539, 7541, 10342, 14184, 18889, 25435, 33974, 44720, 58952, 77550, 100546, 130780, 169273, 217230, 278636, 356566, 452544, 574548, 726938, 914742, 1149685, 1441787, 1798740, 2242436
Offset: 0

Views

Author

N. J. A. Sloane, Nov 27 2000

Keywords

Comments

Convolution inverse is A258941. - Vaclav Kotesovec, Nov 07 2015

Examples

			1 + 7*x + 8*x^2 + 22*x^3 + 42*x^4 + 63*x^5 + 106*x^6 + 190*x^7 + 267*x^8 + ...
T18b = 1/q + 7*q^5 + 8*q^11 + 22*q^17 + 42*q^23 + 63*q^29 + 106*q^35 + ...

Links

Crossrefs

Cf. A030197, A058092, A258941.

Programs

  • Mathematica
    CoefficientList[Series[(QPochhammer[x, x]^3 + 9*x*QPochhammer[x^9, x^9]^3) / (QPochhammer[x, x]*QPochhammer[x^3, x^3]^2), {x, 0, 50}], x] (* Vaclav Kotesovec, Nov 07 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; A:= q^(-1/6)*eta[q]*eta[q^3]^2/(eta[q]^3 + 9*eta[q^9]^3); CoefficientList[Series[1/A, {q, 0, 60}], q] (* G. C. Greubel, Jun 22 2018 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^3 + A) / eta(x + A))^12; polcoeff( ((1 + 27 * x * A)^2 / A)^(1/6), n))} \\ Michael Somos, Jun 16 2012
  • PARI
    q='q+O('q^50); A = (eta(q)^3 + 9*q*eta(q^9)^3)/(eta(q)* eta(q^3)^2); Vec(A) \\ G. C. Greubel, Jun 22 2018

Formula

Expansion of (27 * x * (b(x)^3 + c(x)^3)^2 / (b(x) * c(x))^3)^(1/6) in powers of x where b(), c() are cubic AGM theta functions. - Michael Somos, Jun 16 2012
Expansion of q^(1/6) * a(q) / (b(q) * c(q)/3)^(1/2) in powers of q where a(), b(), c() are cubic AGM theta functions. - Michael Somos, Aug 20 2012
Convolution square is A058092. Convolution sixth power is A030197. - Michael Somos, Jun 16 2012
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Nov 07 2015

A328785 Expansion of q^(-1/3) * (1/3) * b(q)*c(q)/a(q)^2 in powers of q where a(), b(), c() are cubic AGM functions.

Original entry on oeis.org

1, -14, 131, -1080, 8333, -61580, 441893, -3104136, 21454675, -146408362, 988876484, -6622729128, 44039891747, -291092877360, 1914072126008, -12529113820200, 81687913345362, -530724605541430, 3437326269162720, -22199991545327616, 143016156285625823
Offset: 0

Views

Author

Michael Somos, Nov 02 2019

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Convolution square of A258941. Convolution inverse of A058092 with more information there.

Examples

			G.f. = 1 - 14*x + 131*x^2 - 1080*x^3 + 8333*x^4 - 61580*x^5 + ...
G.f. = q - 14*q^4 + 131*q^7 - 1080*q^10 + 8333*q^13 - 61580*q^16 + ...

References

  • B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, 1998.

Crossrefs

Cf. A058092, A258941, A051273.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[x]^2 QPochhammer[x^3]^4 / (QPochhammer[x]^3 + 9 x QPochhammer[x^9]^3)^2, {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); A = ((eta(x + A) * eta(x^3 + A)) / (eta(x^2 + A) * eta(x^6 + A)))^2; polcoeff( 1 / (A + x * 16/A^2), n))};

Formula

Coefficients of 1/3 of power series in equation (13.23), page 179, [Berndt 1998]
A051273(n) = 3*a(n).
a(n) ~ (-1)^n * 64 * Pi^5 * n * exp(Pi*(3*n+1)/3^(3/2)) / Gamma(1/6)^6. - Vaclav Kotesovec, Nov 08 2019
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.