%I #27 Mar 12 2021 22:24:43

%S 1,1,2,4,6,9,14,20,29,42,58,80,110,148,198,264,347,454,592,764,982,

%T 1257,1598,2024,2554,3206,4010,5000,6208,7684,9484,11664,14306,17501,

%U 21346,25972,31526,38170,46112,55588,66861,80258,96154,114968,137212,163472

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

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

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

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 6, 3rd equation.

%H Vaclav Kotesovec, <a href="/A097197/b097197.txt">Table of n, a(n) for n = 0..2000</a>

%H Andrew Sills, <a href="https://works.bepress.com/andrew_sills/40/">Rademacher-Type Formulas for Restricted Partition and Overpartition Functions</a>, Ramanujan Journal, 23 (1-3): 253-264, 2010.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

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

%F Euler transform of period 6 sequence [ 1, 1, 2, 1, 1, 0, ...]. - _Michael Somos_, Aug 19 2006

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

%F 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).

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

%F 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

%e 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 + ...

%e 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 + ...

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

%o (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 */

%Y Cf. A139135.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Sep 17 2004

%E Edited (at the suggestion of _R. J. Mathar_) by _N. J. A. Sloane_, May 15 2008