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

%I A128643 #30 Feb 16 2025 08:33:05
%S A128643 1,9,45,171,549,1566,4095,10008,23157,51201,108918,224100,447831,
%T A128643 872118,1659672,3093498,5658453,10173762,18006021,31408092,54053190,
%U A128643 91869192,154331028,256447080,421789671,687086127,1109128014,1775103507
%N A128643 Expansion of (b(q^2) / b(q))^3 in powers of q where b() is a cubic AGM function.
%C A128643 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A128643 G. C. Greubel, <a href="/A128643/b128643.txt">Table of n, a(n) for n = 0..1000</a>
%H A128643 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A128643 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A128643 Expansion of (chi(-q^3) / chi(-q)^3)^3 in powers of q where chi() is a Ramanujan theta function.
%F A128643 Expansion of ((eta(q^2) / eta(q))^3 * (eta(q^3) / eta(q^6)))^3 in powers of q.
%F A128643 Euler transform of period 6 sequence [ 9, 0, 6, 0, 9, 0, ...].
%F A128643 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.
%F A128643 G.f.: (Product_{k>0} (1 + x^k) / (1 + x^(3*k))^3)^3
%F A128643 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.
%F A128643 a(n) = 9 * A128638(n) unless n = 0. -4*a(n) = A193522(2*n). Convolution inverse of A128642.
%F A128643 a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (8 * 2^(3/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 27 2019
%F A128643 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
%e A128643 1 + 9*q + 45*q^2 + 171*q^3 + 549*q^4 + 1566*q^5 + 4095*q^6 + 10008*q^7 + ...
%t A128643 a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3, q^6] / QPochhammer[ q, q^2]^3)^3, {q, 0, n}]
%o A128643 (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))}
%Y A128643 Cf. A105559, A128638, A128642, A193522.
%K A128643 nonn
%O A128643 0,2
%A A128643 _Michael Somos_, Mar 16 2007