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%I #10 May 22 2014 03:30:28
%S 1,1,0,0,1,1,0,-1,-1,-1,-1,0,0,-2,-1,2,3,0,-1,2,3,0,-1,-1,-2,-2,0,0,
%T -4,-3,5,7,0,-2,4,6,0,-4,-3,-5,-6,0,0,-8,-6,10,14,0,-5,9,13,0,-7,-5,
%U -9,-10,0,0,-16,-12,20,28,0,-8,17,24,0,-14,-11,-18,-20,0,0,-30,-21,36,50,0,-16,30,44,0,-23,-18,-31,-34,0,0,-52,-38,63
%N Expansion of r(q^9) / (r(q) r(q^3)) in powers of q where r() is the Rogers-Ramanujan continued fraction function (A007325).
%C Euler transform of period 45 sequence [1, -1, 0, 1, 0, 0, -1, -1, -1, 0, 1, 0, -1, 1, 0, 1, -1, 1, 1, 0, 0, -1, -1, 0, 0, 1, 1, -1, 1, 0, 1, -1, 0, 1, 0, -1, -1, -1, 0, 0, 1, 0, -1, 1, 0,...].
%F G.f. A(x) satisfies 0=f(A(x),A(x^2)) where f(u,v) = u^2 - v + u*v^3 + u^3*v^2 - 2*u*v*(1 - u + v + u*v).
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w + 2*u*w + 2*u*v*w - u*w^2 - u^2*v -u*v^2*w. - _Michael Somos_, Dec 11 2008
%e q + q^2 + q^5 + q^6 - q^8 - q^9 - q^10 - q^11 - 2*q^14 - q^15 + 2*q^16 + ...
%o (PARI) {a(n)=if(n<1, 0, n--; polcoeff( prod(k=1, n, (1-x^k+x*O(x^n))^-(kronecker(5, k\9-k%9)*if(k%9==0,-1, k%3>0))), n))}
%o (PARI) {a(n)=local(A, u, v); if(n<0, 0, A=x; for(k=2, n, u=A+x*O(x^k); v=subst(u, x, x^2); A-=x^k*polcoeff(u^2-v+u*v^3+u^3*v^2-2*u*v*(1-u+v+u*v), k+1)/2); polcoeff(A, n))}
%K sign
%O 1,14
%A _Michael Somos_, Mar 09 2004, Dec 11 2008
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