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.

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

Offset

0,1

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.

References

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1250

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

Example

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 + ...

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 *)

Prog

(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 */

Crossrefs

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

Sequence in context: A322893 A114943 A119577 * A350901 A160873 A084512

Adjacent sequences: A051270 A051271 A051272 * A051274 A051275 A051276

Keyword

sign

Author

N. J. A. Sloane

Extensions

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

Status

approved