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A228834 Expansion of phi(-x^2)^2 * psi(-x)^4 in powers of x where phi(), psi() are Ramanujan theta functions.

%I A228834 #13 Feb 16 2025 08:33:20
%S A228834 1,-4,2,8,-7,4,-14,-8,18,12,32,-40,-21,-8,-14,32,16,16,-30,56,-14,-28,
%T A228834 -14,-16,-15,-72,66,8,48,52,82,-56,-28,-4,-160,-56,66,84,-32,16,-95,
%U A228834 140,36,56,-30,-112,128,24,-14,-28,-94,-152,64,-156,18,120,98,-80
%N A228834 Expansion of phi(-x^2)^2 * psi(-x)^4 in powers of x where phi(), psi() are Ramanujan theta functions.
%C A228834 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A228834 G. C. Greubel, <a href="/A228834/b228834.txt">Table of n, a(n) for n = 0..1000</a>
%H A228834 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A228834 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A228834 Expansion of q^(-1/2) * (eta(q)^2 * eta(q^4))^2 in powers of q.
%F A228834 Euler transform of period 4 sequence [ -4, -4, -4, -6, ...].
%F A228834 G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 1024 (t / i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A034952.
%F A228834 G.f. (Product_{k>0} (1 - x^k)^2 * (1 - x^(4*k)))^2.
%F A228834 a(2*n) = A228831(n). a(2*n + 1) = -4 * A034952(n).
%e A228834 G.f. = 1 - 4*x + 2*x^2 + 8*x^3 - 7*x^4 + 4*x^5 - 14*x^6 - 8*x^7 + 18*x^8 + ...
%e A228834 G.f. = q - 4*q^3 + 2*q^5 + 8*q^7 - 7*q^9 + 4*q^11 - 14*q^13 - 8*q^15 + 18*q^17 + ...
%t A228834 a[ n_] := SeriesCoefficient[ QPochhammer[ x]^4 * QPochhammer[ x^4]^2, {x, 0 ,n}];
%o A228834 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A))^2, n))};
%Y A228834 Cf. A034952, A228831.
%K A228834 sign
%O A228834 0,2
%A A228834 _Michael Somos_, Sep 04 2013