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A227595 Expansion of phi(-x) * psi(x^3)^2 / chi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

%I A227595 #11 Feb 16 2025 08:33:20
%S A227595 1,-2,0,3,-4,0,4,-2,0,5,-6,0,5,-8,0,5,-2,0,7,-10,0,7,-8,0,9,0,0,7,-12,
%T A227595 0,6,-12,0,11,-6,0,8,-10,0,10,-12,0,8,-4,0,9,-12,0,14,-16,0,10,0,0,15,
%U A227595 -14,0,7,-16,0,7,-8,0,14,-18,0,14,-12,0,16,-2,0,8
%N A227595 Expansion of phi(-x) * psi(x^3)^2 / chi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
%C A227595 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A227595 G. C. Greubel, <a href="/A227595/b227595.txt">Table of n, a(n) for n = 0..1000</a>
%H A227595 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A227595 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A227595 Expansion of q^(-7/8) * eta(q)^2 * eta(q^6)^5 / (eta(q^2) * eta(q^3)^3) in powers of q.
%F A227595 Euler transform of period 6 sequence [ -2, -1, 1, -1, -2, -3, ...].
%F A227595 a(3*n + 2) = a(27*n + 25) = 0.
%e A227595 1 - 2*x + 3*x^3 - 4*x^4 + 4*x^6 - 2*x^7 + 5*x^9 - 6*x^10 + 5*x^12 - 8*x^13 + ...
%e A227595 q^7 - 2*q^15 + 3*q^31 - 4*q^39 + 4*q^55 - 2*q^63 + 5*q^79 - 6*q^87 + 5*q^103 + ...
%t A227595 a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, -x]*EllipticTheta[2, 0, x^(3/2)]^2/(4*x^(3/4)*QPochhammer[x^3, x^6]), {x, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 08 2017 *)
%o A227595 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^5 / (eta(x^2 + A) * eta(x^3 + A)^3), n))}
%K A227595 sign
%O A227595 0,2
%A A227595 _Michael Somos_, Jul 17 2013