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%I #11 Mar 12 2021 22:24:46
%S 1,3,2,8,-5,-4,-10,8,-19,0,14,-16,-10,-4,0,6,14,20,2,0,-11,20,24,-16,
%T 0,-4,14,8,-9,-15,26,0,2,-28,0,-16,-12,-28,-22,0,14,16,0,-30,0,-28,26,
%U 32,-17,0,24,-16,-22,0,-10,32,-34,55,14,0,45,-4,38,8,0,0,-34,-8,38,0,-22,42,2,-28,0,0,-10,20,-48,-40,-20,44
%N Expansion of q^(-1/6) * (eta(q)^4 + 7 * eta(q^7)^4) in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A173030/b173030.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 Expansion of f(-x)^4 + 7 * x * f(-x^7)^4 = chi(-x) * chi(-x^7) * (psi(x)^4 + 7 * x^3 * psi(x^7)^4) in powers of x where psi(), chi(), f() are Ramanujan theta functions.
%F Expansion of (phi(-x)^4 + 7 * phi(-x^7)^4) / (8 * chi(-x) * chi(-x^7)) in powers of x^2 where phi(), chi(), f() are Ramanujan theta functions.
%F a(n) = b(6*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (-p)^(e/2) (1 + (-1)^e)/2 if p == 5 (mod 6).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (252 t)) = 252 (t/i)^2 * f(t) where q = exp(2 Pi i t).
%F A000727(n) = a(n) if n != 1 (mod 7). A000727(7*n + 1) + 7 * A000727(n) = a(7*n + 1).
%e G.f. = 1 + 3*x + 2*x^2 + 8*x^3 - 5*x^4 - 4*x^5 - 10*x^6 + 8*x^7 - 19*x^8 + ...
%e G.f. = q + 3*q^7 + 2*q^13 + 8*q^19 - 5*q^25 - 4*q^31 - 10*q^37 + 8*q^43 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^4 + 7 x QPochhammer[ x^7]^4, {x, 0, n}]; (* _Michael Somos_, Sep 02 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 + 7 * x * eta(x^7 + A)^4, n))};
%Y Cf. A000727.
%K sign
%O 0,2
%A _Michael Somos_, Feb 07 2010
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