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A122859
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Expansion of phi(-q)^3/phi(-q^3) in powers of q where phi() is a Ramanujan theta function.
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3
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1, -6, 12, -6, -6, 0, 12, -12, 12, -6, 0, 0, -6, -12, 24, 0, -6, 0, 12, -12, 0, -12, 0, 0, 12, -6, 24, -6, -12, 0, 0, -12, 12, 0, 0, 0, -6, -12, 24, -12, 0, 0, 24, -12, 0, 0, 0, 0, -6, -18, 12, 0, -12, 0, 12, 0, 24, -12, 0, 0, 0, -12, 24, -12, -6, 0, 0, -12, 0, 0, 0, 0, 12, -12, 24, -6, -12, 0, 24, -12, 0, -6, 0, 0, -12, 0, 24
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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REFERENCES
| N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.64).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of 2*a(q^2)-a(q) = b(q)^2/b(q^2) in powers of q where a(),b() are cubic AGM analog functions.
Expansion of eta(q)^6*eta(q^6)/(eta(q^2)^3*eta(q^3)^2) in powers of q.
Euler transform of period 6 sequence [ -6, -3, -4, -3, -6, -2, ...].
Moebius transform is period 6 sequence [ -6, 18, 0, -18, 6, 0, ...].
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)= v*(u+v)^2 -2*u*w*(v+w).
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=(u1-u2-u3+u6)*(u1+2*u2+u3) -(2*u1+u2-2*u3-u6)*(u1+2*u2-u3).
G.f.: Product_{k>0} (1+x^(3k))/(1+x^k)^3*(1-x^k)^3/(1-x^(3k)) = 1 +6*Sum_{k>0} (-x)^k/(1+x^k+x^(2k)).
a(3n)=a(4n)=a(n). a(6n+5)=0.
G.f.: 1 -6*(Sum_{k>0} x^(3*k-2)/(1+x^(3*k-2)) -x^(3*k-1)/(1+x^(3*k-1))).
(PARI) {a(n)= if(n<1, n==0, 6* sumdiv(n, d, (-1)^(n/d)* kronecker(-3, d)))}
(PARI) {a(n)= if(n<1, n==0, -6* sumdiv(n, d, (2+(-1)^d)* kronecker(-3, d)))}
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^6*eta(x^6+A)/(eta(x^2+A)^3*eta(x^3+A)^2), n))}
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CROSSREFS
| Cf. A113660(n)=(-1)^n*a(n). A122860(n)=-a(n)/6 if n>0.
Sequence in context: A066401 A076590 A113660 * A050496 A103698 A175375
Adjacent sequences: A122856 A122857 A122858 * A122860 A122861 A122862
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KEYWORD
| sign
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AUTHOR
| Michael Somos, Sep 15 2006
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