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A122860
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Expansion of (1-phi(-q)^3/phi(-q^3))/6 in powers of q where phi() is a Ramanujan theta function.
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2
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1, -2, 1, 1, 0, -2, 2, -2, 1, 0, 0, 1, 2, -4, 0, 1, 0, -2, 2, 0, 2, 0, 0, -2, 1, -4, 1, 2, 0, 0, 2, -2, 0, 0, 0, 1, 2, -4, 2, 0, 0, -4, 2, 0, 0, 0, 0, 1, 3, -2, 0, 2, 0, -2, 0, -4, 2, 0, 0, 0, 2, -4, 2, 1, 0, 0, 2, 0, 0, 0, 0, -2, 2, -4, 1, 2, 0, -4, 2, 0, 1, 0, 0, 2, 0, -4, 0, 0, 0, 0, 4, 0, 2, 0, 0, -2, 2, -6, 0, 1, 0, 0, 2, -4, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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 (1+a(q)-2*a(q^2))/6 = (1-b(q)^2/b(q^2))/6 in powers of q where a(),b() are cubic AGM analog functions.
Expansion of (1-eta(q)^6*eta(q^6)/(eta(q^2)^3*eta(q^3)^2))/6 in powers of q.
Moebius transform is period 6 sequence [ 1, -3, 0, 3, -1, 0, ...].
a(n) is multiplicative and a(2^e) = (3(-1)^e-1)/2, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
a(3n)=a(4n)=a(n). a(6n+5)=0.
G.f.: (1-Product_{k>0} (1+x^(3k))/(1+x^k)^3*(1-x^k)^3/(1-x^(3k)))/6 = Sum_{k>0} -(-x)^k/(1+x^k+x^(2k)).
G.f.: 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, 0, -sumdiv(n, d, (-1)^(n/d)* kronecker(-3, d)))}
(PARI) {a(n)= if(n<1, 0, 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( (1-eta(x+A)^6*eta(x^6+A)/(eta(x^2+A)^3*eta(x^3+A)^2))/6, n))}
(PARI) {a(n)=if(n<1, 0, direuler(p=2, n, if(p==2, (1-2*X)/(1-X^2), 1/(1-X)/(1-kronecker(-3, p)*X)))[n])}
(PARI) {a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, (3*(-1)^e-1)/2, if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))}
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CROSSREFS
| Cf. A113661(n)=-(-1)^n*a(n). A122859(n)=-6*a(n) if n>0.
Sequence in context: A113661 A113974 * A123331 A114638 A123340 A110962
Adjacent sequences: A122857 A122858 A122859 * A122861 A122862 A122863
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Sep 15 2006
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