%I A137569
%S A137569 1,1,1,1,1,0,2,1,1,3,2,1,4,3,2,5,4,2,8,6,4,10,7,4,14,10,6,18,13,7,24,17,
%T A137569 10,30,21,12,40,28,17,49,35,19,63,44,26,78,55,31,98,69,40,120,84,47,150,
%U A137569 105,61,182,127,71,224,156,90,271,189,106,330,229,131,396,275
%V A137569 1,-1,-1,1,-1,0,2,-1,-1,3,-2,-1,4,-3,-2,5,-4,-2,8,-6,-4,10,-7,-4,14,-10,
               -6,18,-13,-7,
%W A137569 24,-17,-10,30,-21,-12,40,-28,-17,49,-35,-19,63,-44,-26,78,-55,-31,98,
               -69,-40,120,-84,
%X A137569 -47,150,-105,-61,182,-127,-71,224,-156,-90,271,-189,-106,330,-229,-131,
               396,-275
%N A137569 Expansion of q^(1/12) eta(q) / eta(q^3) in powers of q.
%F A137569 Expansion of f(-q) / f(-q^3) in powers of q where f() is a Ramanujan 
               theta function.
%F A137569 Euler transform of period 3 sequence [ -1, -1, 0, ...].
%F A137569 Given g.f. A(x) then B(x) = A(x^6)^2 / x satisfies 0 = f(B(x), B(x^2), 
               B(x^4)) where f(u, v, w) = 4*v^2 + (u^2 - v) * (w^2 + v).
%F A137569 G.f. is a period 1 Fourier series which satisfies f(-1 / (432 t)) = 3^(1/
               2) / f(t) where q = exp(2 pi i t).`
%F A137569 G.f.: Product_{k>0} (1 - x^(3*k-1)) * (1 - x^(3*k-2)).
%e A137569 q^-1 - q^11 - q^23 + q^35 - q^47 + 2*q^71 - q^83 - q^95 + 3*q^107 + ...
%o A137569 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x 
               + A) / eta(x^3 + A), n))}
%Y A137569 A035943(n) = a(3*n). -A035941(n) = a(3*n+1). -A035940(n) = a(3*n+2). 
               Convolution inverse of A000726.
%Y A137569 Sequence in context: A093394 A094363 A124832 this_sequence A089177 A023996 
               A049998
%Y A137569 Adjacent sequences: A137566 A137567 A137568 this_sequence A137570 A137571 
               A137572
%K A137569 sign
%O A137569 0,7
%A A137569 Michael Somos, Jan 26 2008