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.

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

0,3

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).

References

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 6, 1st equation.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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

Example

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 + ...

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 *)

Prog

(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))};

Crossrefs

Cf. A053268, A153251.

Sequence in context: A350075 A353057 A274406 * A022098 A129659 A041130

Adjacent sequences: A261451 A261452 A261453 * A261455 A261456 A261457

Keyword

nonn

Author

Michael Somos, Nov 18 2015

Status

approved