%I #12 Jul 04 2018 05:51:52

%S 1,-1,-1,2,-2,-1,5,-3,-4,8,-5,-6,16,-8,-11,23,-15,-16,39,-21,-26,58,

%T -35,-39,92,-51,-58,132,-77,-85,194,-108,-125,276,-156,-174,393,-218,

%U -245,542,-304,-336,755,-417,-467,1026,-573,-627,1401,-770,-853,1870

%N Expansion of eta(q) * eta(q^9) * eta(q^21)^2 / (eta(q^3)^2 * eta(q^7) * eta(q^63)) in powers of q.

%H G. C. Greubel, <a href="/A226059/b226059.txt">Table of n, a(n) for n = -1..1500</a>

%F Euler transform of period 63 sequence [ -1, -1, 1, -1, -1, 1, 0, -1, 0, -1, -1, 1, -1, 0, 1, -1, -1, 0, -1, -1, 0, -1, -1, 1, -1, -1, 0, 0, -1, 1, -1, -1, 1, -1, 0, 0, -1, -1, 1, -1, -1, 0, -1, -1, 0, -1, -1, 1, 0, -1, 1, -1, -1, 0, -1, 0, 1, -1, -1, 1, -1, -1, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (u*v - 3) - (u+v) * (u^2 + u*v + v^2).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (63 t)) = 1 / f(t) where q = exp(2 Pi i t).

%e 1/q - 1 - q + 2*q^2 - 2*q^3 - q^4 + 5*q^5 - 3*q^6 - 4*q^7 + 8*q^8 - 5*q^9 + ...

%t eta[q_] := q^(1/24)*QPochhammer[q]; b := eta[q]*eta[q^9]*eta[q^21]^2/ (eta[q^3]^2*eta[q^7]*eta[q^63]); a:= CoefficientList[Series[q*b , {q, 0, 100}], q]; Table[a[[n]], {n, 1, 80}] (* _G. C. Greubel_, Jul 03 2018 *)

%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^9 + A) * eta(x^21 + A)^2 / (eta(x^3 + A)^2 * eta(x^7 + A) * eta(x^63 + A)), n))}

%K sign

%O -1,4

%A _Michael Somos_, May 24 2013