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%I #17 Dec 14 2017 03:46:02
%S 1,-1,-1,0,0,6,-5,-4,0,0,25,-20,-16,0,0,84,-65,-50,0,0,250,-190,-144,
%T 0,0,676,-505,-376,0,0,1706,-1260,-929,0,0,4064,-2970,-2166,0,0,9243,
%U -6700,-4850,0,0,20200,-14535,-10444,0,0,42677,-30520,-21802,0,0,87512,-62235,-44212,0,0,174814
%N Expansion of eta(q)/eta(q^5)^5 in powers of q.
%H Seiichi Manyama, <a href="/A109063/b109063.txt">Table of n, a(n) for n = -1..10000</a>
%H W. Duke, <a href="http://dx.doi.org/10.1090/S0273-0979-05-01047-5">Continued fractions and modular functions</a>, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 150.
%F Euler transform of period 5 sequence [ -1, -1, -1, -1, 4, ...].
%F G.f. A(x) satisfies: 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=u^2*w^2 +2*u*v^2*w +4*u*v^3 -v^3*w.
%F G.f.: (1/x)*Product_{k>0} (1-x^k)/(1-x^(5k))^5.
%F a(5n+2) = a(5n+3) = 0.
%e 1/q -1 -q + 6*q^4 -5*q^5 -4*q^6 + 25*q^9 -20*q^10 -16*q^11 +...
%t QP:= QPochhammer; Table[SeriesCoefficient[QP[q]/QP[q^5]^5, {q, 0, n}], {n, 0, 50}] (* _G. C. Greubel_, Dec 13 2017 *)
%o (PARI) {a(n)=local(A); if(n<-1, 0, n++; A=x*O(x^n); polcoeff( eta(x+A)/eta(x^5+A)^5, n))}
%Y Cf. A053723.
%K sign
%O -1,6
%A _Michael Somos_, Jun 17 2005
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