%I #12 Mar 12 2021 22:24:46

%S 1,-4,14,-36,85,-180,360,-684,1246,-2196,3754,-6264,10226,-16380,

%T 25804,-40032,61275,-92628,138452,-204804,300040,-435672,627356,

%U -896400,1271525,-1791324,2507426,-3488472,4825531,-6638688,9085888,-12373992

%N Expansion of (1/q) * ((chi(q^3) * chi(-q^6)) / (chi(q) * chi(-q^2)))^4 in powers of q where chi() is a Ramanujan theta function.

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

%H G. C. Greubel, <a href="/A193522/b193522.txt">Table of n, a(n) for n = -1..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of - c(-q) * b(q^4) / (b(-q) * c(q^4)) in powers of q where b(), c() are cubic AGM functions.

%F Expansion of (eta(q) * eta(q^4)^2 * eta(q^6)^3 / (eta(q^2)^3 * eta(q^3) * eta(q^12)^2))^4 in powers of q.

%F Euler transform of period 12 sequence [ -4, 8, 0, 0, -4, 0, -4, 0, 0, 8, -4, 0, ...].

%F a(n) = -(-1)^n * A187091(n). a(2*n) = -4 * A128643(n).

%F a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017

%e 1/q - 4 + 14*q - 36*q^2 + 85*q^3 - 180*q^4 + 360*q^5 - 684*q^6 + 1246*q^7 + ...

%t QP := QPochhammer; A193522[n_]:= SeriesCoefficient[((QP[q]*QP[q^4]^2 *QP[q^6]^3)/(QP[q^2]^3*QP[q^3]*QP[q^12]^2))^4, {q, 0, n}];Table[ A193522[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 24 2017 *)

%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^3 / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A)^2))^4, n))}

%Y Cf. A128643, A187091.

%K sign

%O -1,2

%A _Michael Somos_, Jul 29 2011