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A097196 Expansion of psi(x^3)^2 / f(-x^2) in powers of x where psi(), f() are Ramanujan theta functions. 3
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 (list; graph; refs; listen; history; text; internal format)
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.

M. 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

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Last modified September 25 16:45 EDT 2020. Contains 337344 sequences. (Running on oeis4.)