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

Author

N. J. A. Sloane, Sep 17 2004

Keywords

• nonn

Comments

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

Examples

			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 + ...
		

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• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 6, 3rd equation.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

Crossrefs

Cf. A139135.

Programs

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 *)

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 */

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

Extensions

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