A215625 Expansion of q^(2/3) * c(q) / c(q^3) in powers of q where c() is a cubic AGM theta function.

1, 1, 2, -1, 1, -1, 1, -3, -2, 3, 3, 4, -5, 1, -2, 0, -7, -4, 7, 8, 11, -11, 3, -7, 3, -17, -11, 17, 15, 24, -24, 7, -14, 3, -34, -21, 33, 34, 50, -48, 13, -27, 8, -68, -42, 65, 62, 91, -92, 24, -51, 13, -122, -74, 118, 115, 168, -162, 44, -91, 27, -221, -136

Offset

0,3

References

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 349 Entry 2(viii).

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^(2/3) * eta(q^3)^4 / (eta(q) * eta(q^9)^3) in powers of q.

Expansion of f(-q^3)^4 / (f(-q) * f(-q^9)^3) = f(-q^4, -q^5) / f(-q, -q^8) + q * f(-q^2, -q^7) / f(-q^4, -q^5) - q * f(-q, -q^8) / f(-q^2, -q^7) in powers of q where f(), f(,) are Ramanujan theta functions.

Euler transform of period 9 sequence [ 1, 1, -3, 1, 1, -3, 1, 1, 0, ...].

Given g.f. A(x), then B(q) = A(q^3)/q^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^2 * (u * w - v) - u * w * (u + w).

Example

G.f. = 1 + x + 2*x^2 - x^3 + x^4 - x^5 + x^6 - 3*x^7 - 2*x^8 + 3*x^9 + 3*x^10 + ...
G.f. = q^-2 + q + 2*q^4 - q^7 + q^10 - q^13 + q^16 - 3*q^19 - 2*q^22 + 3*q^25 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^4 / (QPochhammer[ x] QPochhammer[ x^9]^3), {x, 0, n}]; (* Michael Somos, Apr 26 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^4 / (eta(x + A) * eta(x^9 + A)^3), n))};
(PARI) q='q+O('q^99); Vec(eta(q^3)^4/(eta(q)*eta(q^9)^3)) \\ Altug Alkan, Mar 30 2018

Crossrefs

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Sequence in context: A249095 A026536 A046213 * A363096 A260222 A181386

Adjacent sequences: A215622 A215623 A215624 * A215626 A215627 A215628

Keyword

sign

Author

Michael Somos, Aug 17 2012

Status

approved