A102315 Expansion of (b(q^6) * c(q^6)) / (b(q^3) * c(q^3)) in powers of q where b(), c() are cubic AGM theta functions.

1, 2, 3, 8, 13, 20, 37, 56, 83, 134, 196, 280, 419, 592, 824, 1176, 1618, 2202, 3040, 4096, 5471, 7368, 9753, 12824, 16937, 22090, 28653, 37248, 47968, 61488, 78887, 100472, 127461, 161702, 203951, 256368, 322090, 402748, 502112, 625464, 776061

Offset: 0

Author

Michael Somos, Jan 04 2005

Keywords

• nonn

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Given g.f. A(x), then B(q) = q*A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v - 4*u*v^2.

Also, B(q) satisfies 0 = f(B(q), B(-q)) where f(u, v) = u + v - 4*u^2*v^2 which is involved in equation (13.22) where gg' = (2B(q))^12 and GG' = (2B(-q))^12. Refer to A058092 for more details. - Michael Somos, Sep 27 2019

Examples

			G.f. = 1 + 2*x + 3*x^2 + 8*x^3 + 13*x^4 + 20*x^5 + 37*x^6 + 56*x^7 + ...
G.f. = q + 2*q^4 + 3*q^7 + 8*q^10 + 13*q^13 + 20*q^16 + 37*q^19 + ...
		

References

• B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag. See p. 179, equation (13.22).
• S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 392.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A058092, A058543.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2] QPochhammer[ x^3, x^6])^-2, {x, 0, n}]; (* Michael Somos, Feb 19 2015 *)
nmax = 60; CoefficientList[Series[Product[(1+x^k)^2 * (1+x^(3*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *)

PARI

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

PARI

q='q+O('q^99); Vec((eta(q^2)*eta(q^6)/(eta(q)*eta(q^3)))^2) \\ Altug Alkan, Apr 21 2018

Formula

Expansion of (chi(-x) * chi(-x^3))^(-2) in powers of x where chi() is a Ramanujan theta function.

Euler transform of period 6 sequence [2, 0, 4, 0, 2, 0, ...].

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

Convolution inverse of A058543. - Michael Somos, Feb 19 2015

a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(11/4)*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 08 2015