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

0,4

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).

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

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

Example

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 + ...

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 *)

Prog

(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 */

Crossrefs

Cf. A053253, A053255.

Sequence in context: A326457 A326543 A326682 * A132325 A308920 A308976

Adjacent sequences: A097193 A097194 A097195 * A097197 A097198 A097199

Keyword

nonn

Author

N. J. A. Sloane, Sep 17 2004

Status

approved