A097196 Expansion of psi(x^3)^2 / f(-x^2) in powers of x where psi(), f() are Ramanujan theta functions.

1, 0, 1, 2, 2, 2, 4, 4, 6, 8, 9, 12, 16, 18, 22, 28, 33, 40, 50, 58, 70, 84, 98, 116, 138, 160, 188, 222, 256, 298, 348, 400, 463, 536, 614, 706, 812, 926, 1060, 1212, 1378, 1568, 1785, 2022, 2292, 2598, 2932, 3312, 3740, 4208, 4736, 5328, 5978, 6708, 7522, 8416, 9416

Offset: 0

Author

N. J. A. Sloane, Sep 17 2004

Keywords

• nonn

Comments

On page 63 of Watson 1936 is an equation with left side 2*rho(q) + omega(q) and the right side is 3 times the g.f. of this sequence. - Michael Somos, Jul 14 2015

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

Examples

			G.f. = 1 + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 6*x^8 + 8*x^9 + ...
G.f. = q^2 + q^8 + 2*q^11 + 2*q^14 + 2*q^17 + 4*q^20 + 4*q^23 + 6*q^26 + ...
		

References

• N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 50, Eq. (25.4).
• George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
• Michael Somos, Introduction to Ramanujan theta functions
• George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A053253, A053255.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(3/2)]^2 / (4 x^(3/4) QPochhammer[ x^2]), {x, 0, n}]; (* Michael Somos, Jul 14 2015 *)
nmax=60; CoefficientList[Series[Product[(1+x^(3*k))^4 * (1-x^(3*k))^2 / (1-x^(2*k)),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 14 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^4 / (eta(x^2 + A) * eta(x^3 + A)^2), n))}; /* Michael Somos, Jul 14 2015 */

Formula

Expansion of q^(-2/3) * eta(x^6)^4 / (eta(x^2) * eta(x^3)^2) in powers of q. - Michael Somos, Jul 14 2015

G.f.: Product_{n >= 1} (1+q^(3*n))^4*(1-q^(3*n))^2/(1-q^(2*n)).

3 * a(n) = A053253(n) + 2 * A053255(n). - Michael Somos, Jul 29 2015

a(n) ~ exp(Pi*sqrt(n/3)) / (12*sqrt(n)). - Vaclav Kotesovec, Oct 14 2015