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A107064 Expansion of q^(-17/24) * (eta(q) * eta(q^6)^4) / (eta(q^2) * eta(q^3)^2) in powers of q.

%I A107064 #20 Feb 16 2025 08:32:57
%S A107064 1,-1,0,1,-1,-1,0,0,0,1,-1,1,0,0,-1,0,1,1,1,0,0,-1,0,-1,-1,1,1,0,0,0,
%T A107064 0,-1,0,-1,1,-1,-1,0,1,-1,1,0,-1,1,0,1,0,0,0,1,-1,-2,0,1,0,0,1,0,1,1,
%U A107064 0,1,0,0,-1,0,-1,0,-2,0,1,1,0,0,0,0,0,0,-1,1,0,0,0,-1,0,-1,1,0,-1,-1,0,0,1,0,-1,-1,0,0,0,1,1,1,0,0,0
%N A107064 Expansion of q^(-17/24) * (eta(q) * eta(q^6)^4) / (eta(q^2) * eta(q^3)^2) in powers of q.
%C A107064 Ramanujan theta functions: f(q) := Product_{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 A107064 G. C. Greubel, <a href="/A107064/b107064.txt">Table of n, a(n) for n = 0..1000</a>
%H A107064 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A107064 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A107064 Euler transform of period 6 sequence [ -1, 0, 1, 0, -1, -2, ...].
%F A107064 G.f.: Product_{k>0} ((1-x^(6k))(1+x^(3k)))^2/(1+x^k).
%F A107064 Expansion of psi(q^3)^2 * chi(-q) in powers of q where psi(), chi() are Ramanujan theta functions.
%F A107064 2*a(n) = - A030204(3*n+2).
%t A107064 eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-17/24)*(eta[q]*eta[q^6]^4)/(eta[q^2]*eta[q^3]^2), {q, 0, 100}], q] (* _G. C. Greubel_, Apr 18 2018 *)
%o A107064 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^4 / eta(x^2 + A) / eta(x^3 + A)^2, n))}
%Y A107064 Cf. A030204.
%K A107064 sign
%O A107064 0,52
%A A107064 _Michael Somos_, May 10 2005