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A228831 Expansion of psi(x)^2 * phi(-x^2)^4 in powers of x where phi(), psi() are Ramanujan theta functions.

%I A228831 #14 Feb 16 2025 08:33:20
%S A228831 1,2,-7,-14,18,32,-21,-14,16,-30,-14,-14,-15,66,48,82,-28,-160,66,-32,
%T A228831 -95,36,-30,128,-14,-94,64,18,98,98,105,-92,-112,-96,-206,-64,-28,226,
%U A228831 -126,-46,320,32,27,-142,208,-30,-60,64,-206,322,-16,-28,-48,-224
%N A228831 Expansion of psi(x)^2 * phi(-x^2)^4 in powers of x where phi(), psi() are Ramanujan theta functions.
%C A228831 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A228831 G. C. Greubel, <a href="/A228831/b228831.txt">Table of n, a(n) for n = 0..2500</a>
%H A228831 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A228831 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A228831 Expansion of f(x)^4 * f(-x)^2 in powers of x where f() is a Ramanujan theta function.
%F A228831 Expansion of q^(-1/4) * (eta(q^2)^6 / (eta(q) * eta(q^4)^2))^2 in powers of q.
%F A228831 Euler transform of period 4 sequence [2, -10, 2, -6, ...].
%F A228831 G.f.: (Product_{k>0} (1 - x^(2*k))^3 * (1 + x^k) / (1 + x^(2*k))^2)^2.
%e A228831 G.f. = 1 + 2*x - 7*x^2 - 14*x^3 + 18*x^4 + 32*x^5 - 21*x^6 - 14*x^7 + 16*x^8 + ...
%e A228831 G.f. = q + 2*q^5 - 7*q^9 - 14*q^13 + 18*q^17 + 32*q^21 - 21*q^25 - 14*q^29 + ...
%t A228831 a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ -x]^4, {x, 0, n}];
%o A228831 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^6 / (eta(x + A) * eta(x^4 + A)^2))^2, n))};
%K A228831 sign
%O A228831 0,2
%A A228831 _Michael Somos_, Sep 04 2013