A097197 Expansion of q^(-1/3) * eta(q^6)^2 / (eta(q) * eta(q^3)) in powers of q.

1, 1, 2, 4, 6, 9, 14, 20, 29, 42, 58, 80, 110, 148, 198, 264, 347, 454, 592, 764, 982, 1257, 1598, 2024, 2554, 3206, 4010, 5000, 6208, 7684, 9484, 11664, 14306, 17501, 21346, 25972, 31526, 38170, 46112, 55588, 66861, 80258, 96154, 114968, 137212, 163472

Offset

0,3

Comments

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

References

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 53, Eq. (25.95).

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 6, 3rd equation.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

a(n) = (-1)^n * A139135(n).

Expansion of psi(q^3) / f(-q) in powers of q where psi(), f() are Ramanujan theta functions.

Euler transform of period 6 sequence [ 1, 1, 2, 1, 1, 0, ...]. - Michael Somos, Aug 19 2006

G.f.: (Sum_{k>=0} x^(3(k^2 + k)/2)) / (Product_{k>0} 1-x^k).

G.f.: (Sum_{k>0} x^(3(k^2 - k)/2)) / ((1 - x) * (1 - x^2) ...) = Product_{k>0} (1 + x^(3*k)) * (1 - x^(6*k)) / (1 - x^k).

G.f.: Product_{k>0} (1 + x^k + x^(2*k)) * (1 + x^(3*k))^2. - Michael Somos, Apr 10 2008

a(n) ~ Pi * BesselI(1, sqrt(6*n+2)*Pi/3) / (2*sqrt(18*n+6)) ~ exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(3/4) * n^(3/4)) * (1 + (-9/(8*Pi) + Pi/3)/sqrt(6*n) + (-5/16 - 45/(256*Pi^2) + Pi^2/108)/n). - Vaclav Kotesovec, Nov 14 2015, extended Jan 09 2017

Example

G.f. = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 + 9*x^5 + 14*x^6 + 20*x^7 + 29*x^8 + 42*x^9 + ...
G.f. = q + q^4 + 2*q^7 + 4*q^10 + 6*q^13 + 9*q^16 + 14*q^19 + 20*q^22 + 29*q^25 + ...

Mathematica

QP = QPochhammer; s=QP[q^6]^2/(QP[q]*QP[q^3]) + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^2 / (eta(x + A) * eta(x^3 + A)), n))}; /* Michael Somos, Aug 19 2006 */

Crossrefs

Cf. A139135.

Sequence in context: A153140 A295341 A139135 * A260600 A119737 A038718

Adjacent sequences: A097194 A097195 A097196 * A097198 A097199 A097200

Keyword

nonn

Author

N. J. A. Sloane, Sep 17 2004

Extensions

Edited (at the suggestion of R. J. Mathar) by N. J. A. Sloane, May 15 2008

Status

approved