A128643 Expansion of (b(q^2) / b(q))^3 in powers of q where b() is a cubic AGM function.

1, 9, 45, 171, 549, 1566, 4095, 10008, 23157, 51201, 108918, 224100, 447831, 872118, 1659672, 3093498, 5658453, 10173762, 18006021, 31408092, 54053190, 91869192, 154331028, 256447080, 421789671, 687086127, 1109128014, 1775103507

Offset: 0

Author

Michael Somos, Mar 16 2007

Keywords

• nonn

Comments

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

Examples

			1 + 9*q + 45*q^2 + 171*q^3 + 549*q^4 + 1566*q^5 + 4095*q^6 + 10008*q^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A105559, A128638, A128642, A193522.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3, q^6] / QPochhammer[ q, q^2]^3)^3, {q, 0, n}]

PARI

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

Formula

Expansion of (chi(-q^3) / chi(-q)^3)^3 in powers of q where chi() is a Ramanujan theta function.

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

Euler transform of period 6 sequence [ 9, 0, 6, 0, 9, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (1 - v) * (1 + 8*u) + (u - v)^2.

G.f.: (Product_{k>0} (1 + x^k) / (1 + x^(3*k))^3)^3

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1 / 8) g(t) where q = exp(2 Pi i t) and g() is g.f. for A105559.

a(n) = 9 * A128638(n) unless n = 0. -4*a(n) = A193522(2*n). Convolution inverse of A128642.

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (8 * 2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 27 2019

Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 1/4 + (1/8)*sqrt(3) + (1/8)*sqrt(9+6*sqrt(3)). - Simon Plouffe, Mar 04 2021