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.

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A153252 Coefficients of the sixth-order mock theta function psi_{-}(q).

Original entry on oeis.org

0, 1, 2, 4, 7, 12, 19, 29, 44, 65, 94, 134, 188, 261, 358, 486, 654, 872, 1155, 1519, 1984, 2576, 3325, 4270, 5456, 6939, 8786, 11077, 13912, 17406, 21700, 26961, 33388, 41221, 50739, 62278, 76232, 93067, 113336, 137684, 166873
Offset: 0

Views

Author

Jeremy Lovejoy, Dec 21 2008

Keywords

Links

Crossrefs

Cf. A153251.
Other '6th-order' mock theta functions are at A053268, A053269, A053270, A053271, A053272, A053273, A053274.

Programs

  • PARI
    lista(nn) =  q = qq + O(qq^nn); gf = sum(n = 1, nn, q^n * prod(k = 1, 2*n-2, 1 + q^k) / prod(k = 1, n, 1 - q^(2*k-1))); concat(0, Vec(gf)) \\Michel Marcus, Jun 18 2013

Formula

G.f.: Sum_{n >= 1} q^n(1+q)(1+q^2)...(1+q^(2n-2))/((1-q)(1-q^3)...(1-q^(2n-1))).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2)*sqrt(3*n)). - Vaclav Kotesovec, Jun 13 2019

A261454 Expansion of a(x^2) / f(-x) in powers of x where a() is a cubic AGM theta function and f() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 8, 9, 17, 25, 47, 63, 106, 144, 216, 296, 425, 569, 807, 1064, 1449, 1905, 2551, 3304, 4353, 5592, 7254, 9247, 11859, 14978, 19038, 23872, 30034, 37433, 46734, 57854, 71739, 88305, 108766, 133191, 163099, 198697, 242069, 293535, 355788, 429609, 518396
Offset: 0

Views

Author

Michael Somos, Nov 18 2015

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x + 8*x^2 + 9*x^3 + 17*x^4 + 25*x^5 + 47*x^6 + 63*x^7 + ...
G.f. = 1/q + q^23 + 8*q^47 + 9*q^71 + 17*q^95 + 25*q^119 + 47*q^143 + ...

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 6, 1st equation.

Links

Crossrefs

Cf. A053268, A153251.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^3 + 9 x^2 QPochhammer[ x^18]^3) / (QPochhammer[ x] QPochhammer[ x^6]), {x, 0, n}];
nmax = 50; CoefficientList[Series[Product[(1 + x^k)^3*(1 - x^k)^2/(1 - x^(6*k)), {k, 1, nmax}] + 9*x^2*Product[(1 - x^(18*k))^3/((1 - x^k)*(1 - x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 15 2019 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 + 9 * x^2 * eta(x^18 + A)^3) / (eta(x + A) * eta(x^6 + A)), n))};

Formula

Expansion of q^(1/24) * (eta(q^2)^3 + 9 * eta(q^18)^3) / (eta(q) * eta(q^6)) in powers of q.
Expansion of phi(x) + 2*phi_{-}(x) in powers of x where phi() and phi_{-}() are 6th-order mock theta functions. [Ramanujan]
a(n) = A053268(n) + 2*A153251(n). [Ramanujan]
a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(3/2)*sqrt(3*n)). - Vaclav Kotesovec, Jun 15 2019
Showing 1-2 of 2 results.

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