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

0,2

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

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

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

Example

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 + ...

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 *)

Prog

(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

Crossrefs

Cf. A058092, A058543.

Sequence in context: A236169 A080478 A002053 * A376534 A142880 A147329

Adjacent sequences: A102312 A102313 A102314 * A102316 A102317 A102318

Keyword

nonn

Author

Michael Somos, Jan 04 2005

Status

approved