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Search: id:A092869
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| A092869 |
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Series expansion of the reciprocal of the Goellnitz-Gordon continued fraction 1 + q + q^2/(1 + q^3 + q^4/(1 + q^5 + q^6/(1 + q^7+ ...))) (cf. A111374). |
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3
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| 1, -1, 0, 1, -1, 1, 0, -2, 2, -1, 0, 2, -3, 2, 0, -2, 4, -4, 0, 4, -6, 5, 0, -6, 9, -6, 0, 7, -12, 9, 0, -10, 16, -13, 0, 15, -22, 17, 0, -20, 29, -21, 0, 25, -38, 28, 0, -32, 50, -39, 0, 43, -64, 49, 0, -56, 82, -60, 0, 69, -105, 78, 0, -86, 132, -101, 0, 112, -166, 125, 0, -142, 208, -153, 0, 172, -258, 192, 0
(list; graph; listen)
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OFFSET
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0,8
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REFERENCES
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S.-D. Chen and S.-S. Huang, On the series expansion of the Goellnitz-Gordon continued fraction, Internat. J. Number Theory, 1 (2005), 53-63.
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc., 42 (2005), 137-162; see Eqs. (9.2),(9.4).
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FORMULA
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Euler transform of period 8 sequence [ -1, 0, 1, 0, 1, 0, -1, 0, ...].
G.f. A(x) satisfies both A(-x)*A(x) = A(x^2) and xA(x)^2 = B(xA(x^2)) where B(x) = x*(1-x)/(1+x).
Ramanujan's theta functions psi(q)=f(q, q^3)=A010054(q), phi(q)=f(q, q)=theta_3(q)=A000122(q).
G.f.: Product_{k>=0} (1-x^(8*k+1))*(1-x^(8*k+7))/((1-x^(8*k+3))(1-x^(8*k+5))).
Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x), B(x^2)) where f(u, v)=u^2-v+v^2+v*u^2.
Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x), B(x^3)) where f(u, v)=(1-u*v)*(u+v)^3 -(v+v^3)*(1-u^4). - Michael Somos Feb 15 2006
Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x), B(x^5)) where f(u, v)=(u-v)*(1+u*v)^5 -u*(1-u^4)*(1+v^2)*(1-6*v^2+v^4). - Michael Somos Feb 15 2006
Expansion of (phi(q)-phi(q^2))/(2*q*psi(q^4)) = 2*psi(q^4)/(phi(q)+phi(q^2)) in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos Feb 15 2006
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EXAMPLE
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q -q^3 +q^7 -q^9 +q^11 -2*q^15 +2*q^17 -q^19 +2*q^23 +...
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PROGRAM
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(PARI) a(n)=local(A, u, v); if(n<0, 0, n=2*n+1; A=x; forstep(k=3, n, 2, u=A+x*O(x^k); v=subst(u, x, x^2); A-=x^k*polcoeff(u^2-v+v*u^2+v^2, k+1)/2); polcoeff(A, n))
(PARI) a(n)=local(A, m); if(n<0, 0, A=1+O(x); m=1; while(m<=n, m*=2; A=x*subst(A, x, x^2); A=sqrt(A*(1-A)/(1+A)/x)); polcoeff(A, n))
(PARI) a(n)=local(A, A2); if(n<0, 0, A=eta(x^8+x*O(x^n))^2/eta(x^4+x*O(x^n)); A2=sum(k=1, sqrtint(n), x^k^2+x^(2*k^2), 1+x*O(x^n)); polcoeff(A/A2, n))
(PARI) {a(n)=local(A, A2); if(n<0, 0, A=x*O(x^n); A=eta(x+A)*eta(x^4+A)^2/eta(x^2+A)^3; A2=subst(A, x, x^2); polcoeff(2*A^2*A2^2/(A^2+A2), n))}
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CROSSREFS
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A083365(n)=a(4n). Cf. A091188, A111374, A003823.
Sequence in context: A001617 A084934 A125927 this_sequence A029337 A060086 A062135
Adjacent sequences: A092866 A092867 A092868 this_sequence A092870 A092871 A092872
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Mar 07 2004; corrected Jun 09 2004
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