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Search: id:A113974
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| A113974 |
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Expansion of (1-phi(x^3)^3/phi(x))/2 where phi() is a Ramanujan theta function. |
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3
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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; listen)
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OFFSET
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1,2
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 375 Entry 35.
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FORMULA
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a(n) is multiplicative and a(2^e) = (1-3(-1)^e)/2 if e>0, 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).
Moebius transform is period 12 sequence [1, -3, 0, 1, -1, 0, 1, -1, 0, 3, -1, 0, ...].
G.f.: (1-theta_3(q^3)^3/theta_3(q))/2.
G.f.: Sum_{k>0} x^(3k-2)/(1-(-x)^(3k-2)) - x^(3k-1)/(1-(-x)^(3k-1)) = Sum_{k>0} -(-1)^k x^k/(1+x^k+x^(2k)) -2 x^(4k)/(1+x^(4k)+x^(8k)).
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PROGRAM
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(PARI) {a(n)=local(x); if(n<1, 0, x=valuation(n, 2); if(n%2, 1, (-3+(-1)^x)/2)*sumdiv(n/2^x, d, kronecker(-3, d)))}
(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)/2, if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))}
(PARI) {a(n)=if(n<1, 0, direuler(p=2, n, if(p==2, 2-(1+2*X)/(1-X^2), 1/(1-X)/(1-kronecker(-3, p)*X)))[n])}
(PARI) {a(n)=local(A); if(n<0, 0, A=sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n)); polcoeff( (1-subst(A+x*O(x^(n\3)), x, x^3)^3/A)/2, n))}
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CROSSREFS
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A113973(n)=-2*a(n) if n>0.
Adjacent sequences: A113971 A113972 A113973 this_sequence A113975 A113976 A113977
Sequence in context: A025860 A058394 A113661 this_sequence A122860 A123331 A114638
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KEYWORD
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sign,mult
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AUTHOR
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Michael Somos, Nov 10 2005
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