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 A109389 Expansion of q^(-1/12)eta(q)eta(q^6)/(eta(q^2)eta(q^3)) in powers of q. 0

%I

%S 1,-1,0,0,0,-1,1,-1,1,0,0,-1,2,-2,1,0,1,-2,3,-3,2,-1,1,-3,5,-5,3,-1,2,

%T -5,7,-7,5,-3,3,-7,11,-11,7,-4,6,-11,15,-15,11,-7,8,-15,22,-22,15,-10,

%U 13,-22,30,-30,23,-16,18,-30,42,-42,31,-22,27,-43,56,-56,44,-33,37,-57,77,-77,59,-45,53,-79,101,-101,82,-64

%N Expansion of q^(-1/12)eta(q)eta(q^6)/(eta(q^2)eta(q^3)) in powers of q.

%F Euler transform of period 6 sequence [ -1, 0, 0, 0, -1, 0, ...].

%F G.f.: 1/(Product_{k>0} (1+^(2k-1)+x^(4k-2))) = Product_{k>0} (1-x^(6k-1))(1-x^(6k-5)) = Product_{k>0} (1-x^k+x^(2k)) (where 1-x+x^2 is 6th cyclotomic polynomial).

%F Given g.f. A(x), then B(x)=x*A(x^12) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=(v^2+u^4)*(v^2+w^4)-2*v^4*(1-v*u^2*w^2).

%F Expansion of G(x^6) * H(x) - x * G(x) * H(x^6) where G(), H() are Rogers-Ramanujan functions.

%e q - q^13 - q^61 + q^73 - q^85 + q^97 - q^133 + 2*q^145 - 2*q^157 + q^169 + ...

%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^6+A)/eta(x^2+A)/eta(x^3+A), n))}

%Y Cf. a(n)=(-1)^n*A098884(n).

%K sign

%O 0,13

%A Michael Somos, Jun 26 2005

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