login
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
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
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
Sequence in context: A326457 A326543 A326682 * A132325 A308920 A308976
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Sep 17 2004
STATUS
approved