login
Expansion of q^(-1/24) * (eta(q^2) * eta(q^3)^4) / (eta(q) * eta(q^6)^2) in powers of q.
2

%I #20 Mar 12 2021 22:24:43

%S 1,1,1,-2,-2,-1,0,1,-2,0,-2,0,3,2,2,-1,0,2,-2,2,0,0,1,0,2,-2,1,0,-2,

%T -4,0,0,-2,0,0,1,0,0,0,-2,1,0,-2,-2,0,0,0,2,2,0,2,1,2,0,-2,2,0,1,0,0,

%U 0,0,-2,4,0,0,0,-2,0,2,3,0,0,-2,2,-2,-2,-1,-2,0,-4,0,0,2,-2,0,0,-2,2,2,-2,0,1,0,0,-2,0,-4,0,2,1,-2,0,-2,0

%N Expansion of q^(-1/24) * (eta(q^2) * eta(q^3)^4) / (eta(q) * eta(q^6)^2) in powers of q.

%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

%H G. C. Greubel, <a href="/A107063/b107063.txt">Table of n, a(n) for n = 0..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 Euler transform of period 6 sequence [1, 0, -3, 0, 1, -2, ...].

%F G.f.: Product_{k>0} (1+x^k)*(1-x^(3*k))^2/(1+x^(3*k))^2.

%F Expansion of phi(-q^3)^2 / chi(-q) in powers of q where phi(), chi() are Ramanujan theta functions.

%t eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/24)* (eta[q^2]*eta[q^3]^4)/(eta[q]*eta[q^6]^2), {q, 0, 100}], q] (* _G. C. Greubel_, Apr 18 2018 *)

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

%Y A030204(3*n) = a(n).

%K sign

%O 0,4

%A _Michael Somos_, May 10 2005